MOX-Report No. 23/2019
Derivation and analysis of coupled PDEs on manifoldswith high dimensionality gap arising from topological
model reduction
Laurino, F; Zunino, P.
MOX, Dipartimento di Matematica Politecnico di Milano, Via Bonardi 9 - 20133 Milano (Italy)
[email protected] http://mox.polimi.it
Mathematical Modelling and Numerical Analysis Will be set by the publisher
Modelisation Mathematique et Analyse Numerique
DERIVATION AND ANALYSIS OF COUPLED PDES ON MANIFOLDS WITH
HIGH DIMENSIONALITY GAP ARISING FROM TOPOLOGICAL MODEL
REDUCTION ∗
Federica Laurino1, 2 and Paolo Zunino1
Abstract. Multiscale methods based on coupled partial differential equations defined on bulkand embedded manifolds are still poorly explored from the theoretical standpoint, although theyare successfully used in applications, such as microcirculation and flow in perforated subsurfacereservoirs. This work aims at shedding light on some theoretical aspects of a multiscale methodconsisting of coupled partial differential equations defined on one-dimensional domains embeddedinto three-dimensional ones. Mathematical issues arise because the dimensionality gap between thebulk and the inclusions is larger than one, that is the high dimensionality gap case. First, we showthat such model derives from a system of fully three-dimensional equations, by the application ofa topological model reduction approach. Secondly, we rigorously analyze the problem, showingthat the averaging operators applied for the model reduction introduce a regularization effect thatresolves the issues due to the singularity of solutions and to the ill-posedness of restriction operators.Then, we exploit the structure of the model reduction technique to analyze the modeling error. Thisstudy confirms that for infinitesimally small inclusions, the modeling error vanishes. Finally, wediscretize the problem by means of the finite element method and we analyze the approximationand the model error by means of numerical experiments.
1991 Mathematics Subject Classification. 35J25, 58J05, 58C05, 65N30, 65N15, 65N85.
The dates will be set by the publisher.
1. Introduction
The topological (or geometrical) model reduction techniques play an essential role in the simulation ofmultiscale, multiphysics, multimodel systems. For example, small inclusions of a continuum can be describedas zero-dimensional (0D) or one-dimensional (1D) concentrated sources in order to reduce the computationalcost of simulations. Many problems in this area are not well investigated yet, such as the coupling of three-dimensional (3D) continua with embedded (1D) networks, although it arises in applications of paramountimportance. For example, such models have been introduced since three decades (at least), for modelingwells in subsurface reservoirs in [35,36] and for modeling microcirculation in [5,17,18,42]. A similar approachhas been recently used to model soil/root interactions [23]. However, these application-driven seminal ideaswere not followed by a systematic theory and rigorous mathematical analysis.
Keywords and phrases: embedded geometric multiscale, heterogeneous domain dimensionality, model error analysis;
∗ The authors are members of the INdAM Research group GNCS. The author Federica Laurino acknowledges the supportof the Italian Institute of Technology with the fellowship: Sviluppo di metodi computazionali di tipo multi-scala per la
simulazione del trasporto vascolare ed extra vascolare di molecole, nano-costrutti e cellule in tessuti neoplastici.1 MOX, Department of Mathematics, Politecnico di Milano;
e-mail: [email protected] & [email protected] Italian Institute of Technology, Genova, Italy
c© EDP Sciences, SMAI 1999
Seminal works featuring the formulation and analysis of bulk 3D with embedded 1D PDEs, due toD’Angelo and co-workers [12–14, 30], have appeared much later than their applications. From the pointof view of numerical approximation and analysis of such problems, besides the works by D’Angelo, we men-tion [4, 6, 24–28, 33]. More precisely, in [4, 24, 25] optimal a-priori error estimates for the finite elementapproximation of elliptic equations in with Dirac sources are addressed. The solution of 1D differentialequations embedded in 2D is studied in [27], recently extended to 3D-1D in [28]. The consistent derivationof numerical approximations schemes for PDEs in mixed dimension is addressed in [6], while [33] focuses onthe approximation of 3D-1D coupled problems with mixed finite elements. Relevant topics related to the3D-1D coupling have also been addressed in [21,44], where regularization of singularities is studied for PDEswith singular source terms.
We believe that these studies do not cover the many facets of a comprehensive theory of coupled PDEson embedded manifolds with heterogeneous dimensionality. The main issues consist in the poor regularityof solutions of (second order, elliptic) PDEs with singular forcing terms, combined with the ill-posednessof restriction operators (such as the trace operator) applied on manifolds with co-dimension larger thanone. We envision different ways to overcome this theoretical obstacle. One is the application of weightedSobolev spaces to handle singular functions, already adopted in [13,14]. The other one is the introduction ofregularizing operators. This work explores the latter path, with the main purpose to extend the work of [26]to the more challenging case of 3D-1D coupled equations.
We start from a system of fully 3D second order elliptic equations, coupled by means of Robin type (ormixed type) interface conditions that mimic the flux type interface conditions that are typically used in theapplications cited above. This represents an idealized problem of transport/flow through a network of smallinclusions embedded into a bulk domain. We identify three general assumptions that allow us to transformthe original problem into a simpler one, by means of a topological model reduction technique based on localaveraging. The resulting equations consist of 3D-1D coupled elliptic equations. Since the mathematicaltools used for the derivation of the reduced model are related to the geometric multiscale method, see forexample [39] for a recent review, but are applied here to model small inclusions embedded into a bulk, wecall this new approach the geometric embedded multiscale method.
We show that the averaging operators introduce in the problem a regularizing effect, such that theweak solution exists in classical Hilbert spaces. As a consequence, the reduced problem can be naturallyapproximated by means of finite elements and, thanks to some additional regularity results, the convergence ofthe approximation method can be proved and the corresponding rates are observed in numerical experiments.We notice that if the small inclusions are shaped as a network, the 1D subproblem consists of a quantumgraph, see [3] for details, and its finite element approximation can be addressed with the tools recentlydeveloped in [2]. Furthermore, the systematic derivation of the reduced model reveals the structure of themodeling error, allowing us to perform a rigorous analysis of it. This analysis ends up with the conclusionthat for infinitesimally small inclusions, the reduced model is equivalent to the fully 3D one.
We believe that building sound mathematical foundations for the 3D-1D approach to model embeddedinclusions will benefit to the theoretical knowledge and it will also improve the reliability of the method forrelevant applications, as for example the ones already addressed by the authors and co-workers in [9, 10,31,32,37,38], see Figure 1 (right panel) for an illustration of these results.
2. Problem setting
We address here the geometrical configuration of the domain and we set up a problem based on Robin-Neumann coupling conditions. Figure 1 (left panel) shows a qualitative sketch of the problem setting. Then,we present the formal derivation of the reduced problem, consisting of 3D-1D coupled equations. A rigorousanalysis of it will be considered in Section 3.
2.1. Geometrical setting
The domain is denoted as Ω and composed by two parts, Σ and Ω⊕ := Ω \ Σ. We assume Ω is convexand Σ is completely embedded into Ω, such that the distance between ∂Ω and ∂Σ is strictly positive.
Let Σ be a generalized cylinder, that is the swept volume of a two dimensional set moved along a curve inthe three-dimensional space, see for example [19]. More precisely, let λ(s) = [ξ(s), τ(s), ζ(s)], s ∈ (0, S) be a
2
Figure 1. The left panel shows a qualitative sketch of the geometrical setting of the prob-lem. The right panel illustrates three significant applications for which the geometric em-bedded multiscale method has been already successfully used: case (A) represents micro-circulation, addressed in [9, 10, 31, 32, 37, 38]; case (B) shows a preliminary study of theinteraction of soil and roots pursued in [40]; case (C) represents a prototype simulation forthe interaction of reservoirs and wells performed with the approach described in [11].
C2-regular curve in the three-dimensional space. Let Λ = λ(s), s ∈ (0, S) be the centerline of the cylinder.For simplicity, let us assume that ‖λ′(s)‖ = 1 such that the arc-length and the coordinate s coincide. LetT ,N ,B be the Frenet frame related to the curve.
Let D(s) = [x(r, t), y(r, t)] : (0, R(s)) × (0, T (s)) → R2 be a parametrization of the cross section. Let usassume that D(s) is convex for any s ∈ (0, S). The cross section can change size along Λ but not shape. Let usalso parametrize the boundary of the cross section as ∂D(s) = [∂x(r, t), ∂y(r, t)] : (0, R(s))× (0, T (s))→ R2,and let us assume that ∂D(s) is a piecewise C2-regular curve. Then, the generalized cylinder Σ can be definedas follows
Σ = λ(s) + x(r, t)N(s) + y(r, t)B(s), r ∈ (0, R(s)), s ∈ (0, S), t ∈ (0, T (s)) ,and the lateral boundary of it, denoted with Γ, is
Γ = λ(s) + ∂x(r, t)N(s) + ∂y(r, t)B(s), r ∈ (0, R(s)), s ∈ (0, S), t ∈ (0, T (s)) .
Let | · | denote the Lebesgue measure of a set. If |D(0)|, |D(S)| > 0 then Σ has top and bottom boundaries,which are ∂Σ \Γ = λ(0) +D(0) ∪ λ(S) +D(S). For simplicity of notation we write Γ0 = λ(0) +D(0)and ΓS = λ(S) +D(S).
Thanks to the geometrical structure of the generalized cylinder, we decompose integrals as follows, forany sufficiently regular function w,∫
Σ
wdω =
∫Λ
∫D(s)
wdσds =
∫Λ
|D(s)|w(s)ds , where w(s) = |D(s)|−1
∫D(s)
wdσ ,∫∂Σ
wdσ =
∫Λ
∫∂D(s)
wdγds =
∫Λ
|∂D(s)|w(s)ds , where w(s) = |∂D(s)|−1
∫∂D(s)
wdγ ,
being dω, dσ, dγ the generic volume, surface and curvilinear Lebesgue measures.With little abuse of notation, for a straight cylinder we identify the function w(s) : Λ → R with the
function on Σ obtained by extending the mean value to each cross section D(s). The same extension can bealso applied to Γ, namely w(s) can be either regarded as a function on Λ or on Γ.
Let us now formulate a fundamental assumption (A0) on the proportions of Σ.3
A0: We assume that the transversal diameter of Σ is small compared to the diameter of Ω. The smallparameter (defined below) is denoted with the symbol ε.
Let D = maxs∈(0,S) diam(D(s)) be the largest diameter of the cross sections of Σ, that is the transversaldiameter of Σ. The central assumption of this work is that D diam(Ω). Let us now scale the domains Ωand Σ. Let χΩ(x) = x/diam(Ω) be a scaling function and let be Ωχ = χΩ(Ω), Σχ = χΩ(Σ) be the scaleddomains. The previous assumption implies that for the scaled domains ε = Dχ = D/diam(Ω) is such that0 < ε 1. For simplicity of notation, and without loss of generality, form now on we will implicitly refer tothe scaled domains dropping the subindex χ.
2.2. Fully 3D coupled problem formulation
We describe here a version of the problem arising from Robin-Neumann conditions. After straightforwardlyrearranging the Robin-Neumann conditions as Robin-Robin ones, the problem consists to find u⊕, u (where⊕, denote the exterior and the interior of Σ, respectively) such as:
−∆u⊕ = f in Ω⊕, (1a)
−∆u = g in Σ, (1b)
−∇u⊕ · n⊕ = κ (u⊕ − u) on Γ, (1c)
−∇u · n = κ (u − u⊕) on Γ, (1d)
−∇u⊕ · n⊕ = 0 on Γ0 ∪ ΓS , (1e)
−∇u · n = 0 on Γ0 ∪ ΓS , (1f)
u⊕ = 0 on ∂Ω . (1g)
It is assumed that the interface of Σ is permeable, namely it is crossed by a normal flux proportional toκ (u⊕ − u). The coefficient κ plays the role of permeability or transfer coefficient and it assumes a uniformvalue on each cross section ∂D(s). As a result of that, κ is only a (regular) function of the arc-length s.For the boundary conditions on the top and bottom faces of the cylinder, we make the assumption that|D(0)|, |D(S)| > 0. In applications, the domain Σ consists of many cylinders, representing channels carryingflow, fibers or inclusions. The numerical approximation of PDEs on such domain may thus become verydemanding, due to its complex shape. The main disadvantage is that it requires the resolution of the fullgeometry of the inclusions, which in many real applications can be difficult to handle. For this reason, weaim to apply topological model reduction techniques, based on averaging, in order to transform the problemon Σ into a simpler one.
The objective of this work is to consider a simplified version of problem (1), where the domain Σ shrinks toits centerline Λ and the corresponding partial differential equation is averaged on the cylinder cross section,namely D. This new problem setting will be called the reduced problem. From the mathematical standpointit is more challenging than (1), because it involves the coupling of 3D/1D elliptic problems.
2.3. Topological model reduction of the problem on Σ
We apply the averaging technique to equation (1b). In particular, we consider an arbitrary portion P ofthe cylinder, with lateral surface ΓP and bounded by two perpendicular sections to Λ, namely D(s1), D(s2)with s1 < s2. We have,∫
P∆udω =
∫∂P∇u · n dσ = −
∫D(s1)
∂sudσ +
∫D(s2)
∂sudσ +
∫ΓP
∇u · ndσ .
By the fundamental theorem of integral calculus combined with the Reynolds transport Theorem, we have
−∫D(s1)
∂sudσ +
∫D(s2)
∂sudσ =
∫ s2
s1
ds
∫D(s)
∂sudσds
=
∫ s2
s1
d2ss
∫D(s)
u dσ ds−∫ s2
s1
ds
(∫∂D(s)
νu dγ
)ds ,
4
where ν denotes the normal deformation of the boundary along (0, S). More precisely, we observe that ν isuniform on ∂D(s), because we assume that D(s) can not change shape. Then, the following identity holdstrue,
ds(|D(s)|) =
∫∂D(s)
ν dγ = |∂D(s)|ν. (2)
Substituting in the previous equality, we obtain∫ s2
s1
d2ss
∫D(s)
u dσ ds−∫ s2
s1
ds
(∫∂D(s)
νu dγ
)ds =
∫ s2
s1
[d2ss(|D(s)|u)− ds(ν|∂D(s)|u)
]ds
=
∫ s2
s1
[d2ss(|D(s)|u)− ds (ds(|D(s)|)u)
]ds.
By means of (1d) we obtain,∫ΓP
∇u · ndσ = −∫
ΓP
κ(u − u⊕) dσ = −∫ s2
s1
∫∂D(s)
κ(u − u⊕)dγ ds = −∫ s2
s1
κ|∂D(s)|(u − u⊕) ds .
From the combination of all the above terms with the right hand side, we obtain that the solution u of (1)satisfies, ∫ s2
s1
[−d2
ss(|D(s)|u) + ds (ds(|D(s)|)u) + |∂D(s)|κ(u − u⊕)]ds =
∫ s2
s1
|D(s)|g ds .
Since the choice of the points s1, s2 is arbitrary, we conclude that the following equation holds true,
−d2ss(|D(s)|u) + ds (ds(|D(s)|)u) + |∂D(s)|κ(u − u⊕) = |D(s)|g on Λ , (3)
which is complemented by the following conditions at the boundary of Λ,
|D(s)|dsu = 0, ds|D(s)| = 0, on s = 0, S. (4)
Then, we consider the variational formulation of the averaged equation (3). After multiplication by a testfunction V ∈ H1(Λ), integration on Λ and suitable application of integration by parts, we obtain,∫
Λ
ds(|D(s)|u)dsV ds− ds(|D(s)|u)V |s=Ss=0 −∫
Λ
(ds|D(s)|)udsV ds+ (ds|D(s)|)uV |s=Ss=0
+
∫Λ
|∂D(s)|κ(u − u⊕)V ds =
∫Λ
|D(s)|gV ds .
Using boundary conditions, the identity ds(|D(s)|u) = |D(s)|dsu + ds(|D(s)|)u and reminding thatds|D(s)|)/|∂D(s)| = ν, we obtain,
(dsu, dsV )Λ,|D| + (ν(u − u), dsV )Λ,|∂D| + (κ(u − u⊕), V )Λ,|∂D| = (g, V )Λ,|D| . (5)
where we have introduced the following weighted inner product notation,
(U, V )Λ,w =
∫ S
0
w(s)U(s)V (s)ds .
Let us now formulate the modelling assumption that allows us to reduce equation (5) to a solvable one-dimensional (1D) model. More precisely, we assume that:
A1: the function u has a uniform profile on each cross section D(s), namely u(r, s, t) = U(s).5
Therefore, observing that U = U = U , problem (5) consists to find U ∈ H1(Λ) such that
(dsU, dsV )Λ,|D| + (κU, V )Λ,|∂D| = (κu⊕, V )Λ,|∂D| + (g, V )Λ,|D| ∀V ∈ H1(Λ) . (6)
Remark 2.1. By formally writing the strong formulation of equation (6) we obtain,
−d2ssU +
|∂D||D|
κU =|∂D||D|
κu⊕ + g, on Λ .
We observe that the relative magnitude of the flux terms, that are (|∂D|/|D|)κ(U−u⊕), scales as (|∂D|/|D|)κwith respect to the volumetric terms, namely d2
ssU + g. This sheds light on the behavior of the model whenε→ 0. More precisely, for a bounded κ, what matters is the limit of |∂D|/|D|. Assuming that ∂D is rectifiableand the cross section D is characterized by two axes, of length ε and εα respectively (where the parameterα controls the aspect ratio of the cross section), we have limε→0 |∂D|/|D| ' ε−α + ε−1 ' ε−max(α,1). Unlessα > 1, the scaling of |∂D|/|D| is ε−1. Going back to the strong formulation of (6), this entails that the fluxterms diverge when ε → 0. This observation explains, from an heuristic yet physical standpoint, why thecoupling between the 3D and the 1D models becomes ill posed when the radius of the 1D inclusion vanishes.
2.4. Topological model reduction of the problem on Ω⊕
We focus here on the subproblem of (1) related to Ω⊕, that is
−∆u⊕ = f in Ω⊕, (7a)
−∇u⊕ · n⊕ = κ (u⊕ − u) on Γ, (7b)
u⊕ = 0 on ∂Ω . (7c)
First, we multiply both sides of (7a) for a test function v ∈ H10 (Ω),
−∫
Ω⊕
∆u⊕v dω =
∫Ω⊕
fv dω.
Integrating by parts and using boundary and interface conditions, we obtain:∫Ω⊕
fv dω = −∫
Ω⊕
∆u⊕v dω =
∫Ω⊕
∇u⊕ · ∇v dω −∫∂Ω⊕
∇u⊕ · n⊕v dσ
=
∫Ω⊕
∇u⊕ · ∇v dΩ +
∫Γ
κ(u⊕ − u)v dσ .
Now, we apply a topological model reduction of the interface conditions, namely we go from a 3D-3D toa 3D-1D formulation. To this purpose, let us write the solution and the test functions on every cross section∂D(s) as their average plus some fluctuation,
u⊕ = u⊕ + u⊕, u = u + u, v = v + v, on ∂D(s) ,
where u⊕ = u = v = 0. Therefore, using the coordinates system (r, s, t) on Γ, for ∗ = ⊕, we have,∫Γ
κu∗v dσ =
∫Λ
κ
∫∂D(s)
(u∗ + u∗)(v + v)dγds =
∫Λ
κ|∂D(s)|u∗v ds+
∫Λ
κ
∫∂D(s)
u∗vdγds .
Then, we make the following modelling assumptions:
A2: we identify the domain Ω⊕ with the entire Ω, and we correspondingly omit the subscript ⊕ to thefunctions defined on Ω⊕, namely ∫
Ω⊕
u⊕ dω '∫
Ω
u dω .
6
A3: we assume that the product of fluctuations is small, namely
∫∂D(s)
u∗vdγ ' 0 .
By means of the previous deductions, reminding that for assumption A1 we have that u = U and puttingtogether the terms of the weak form of (7), we obtain that u solves the following problem,
(∇u,∇v)Ω + (κu, v)Λ,|∂D| = (κU, v)Λ,|∂D| + (f, v)Ω , ∀v ∈ H10 (Ω) . (8)
2.5. Extension of the 1D problem to a metric graph
The embedded domain Σ was defined starting from its centerline, namely the curve Λ. We now discussthe generalization to the case where Λ is a network. In our case, the edges of the network are curvesλi(si) = [ξi(si), τi(si), ζi(si)], si ∈ (0, Si), i = 1, . . . , N that are connected at a number M of vertices,
yj = λi(0) = λı(Sı), i, ı ∈ 1, . . . , N, j = 1, . . . ,M.
The set of vertices is denoted with Y = yj ∈ Rd, j = 1, . . . ,M, while Kj , represent all the indices
i that are connected with the vertex j. Furthermore, Kj can be decomposed into K−j and K+j , where
K−j = i ∈ 1, . . . , N : yj = λi(0) are the branches originating in the j−th vertex, according to their
orientation. The complementary is K+j denoting the branches that end into the same vertex. We denote with
i ∈ B the indices of segments with a dead-end, which can be similarly split into B+, B−. Obviously, eachedge has an arc-coordinate si and a length Si, under the assumption ‖λ′i‖ = 1. With these two properties,the network is also a metric graph. In this more general setting the embedded domain Σ is defined as theunion of all the generalized cylinders generated by swiping suitable sections ∂D(si) along the centerlines
Λi = λi(si), si ∈ (0, Si), namely Σ =⋃Ni=1 Σi.
We observe that the topological model reduction approach can still be applied branch by branch indi-vidually, but it can not be adapted to the entire Σ at once, because in proximity of the junctions Σ is nolonger a generalized cylinder. For this reason we define the reduced problem directly from the differentialformulation of a single branch. Equation (3) applies to each edge Λi and it must be complemented withsuitable matching conditions at the vertices, which correspond to the application of (4) at the endpoints ofeach Λi. Such conditions turn out to be the Kirchhoff ones that for the j-th vertex can be written as,
∑i∈K+
j
|D(Si)|dsiUi(Si)−∑i∈K−j
|D(0)|dsiUi(0) = 0 ,
Ui(0) = Uı(Sı) , ∀i ∈ K−j , ı ∈ K+j ,
dsi |D(0)| = dsi |D(Si)| = 0 , ∀i ∈ Kj .
The first conditions corresponds to balance of current or fluxes, while the second states that the solution oneach edge must be continuous at the vertices. The third condition is just an assumption on the shape ofeach branch at the endpoints. The reduced problem on the network consists to find a collection of functionsUi, i = 1, . . . , N such that
7
−d2ss(|D(si)|Ui) + dsi(dsi |D(si)|Ui) + |∂D(si)|κiUi (9a)
=|∂D(si)|κiu⊕ + |D(si)|g , on Λi, ∀i = 1, . . . , N ,
dsi |D(0)| = dsi |D(Si)| = 0 , ∀i ∈ Kj , ∀j = 1, . . . ,M , (9b)∑i∈K+
j
|D(Si)|dsiUi(Si)−∑i∈K−j
|D(0)|dsiUi(0) = 0 , ∀j = 1, . . . ,M , (9c)
Ui(0) = Uı(Sı) , ∀i ∈ K−j , ∀ı ∈ K+j , ∀j = 1, . . . ,M , (9d)
|D(0)|dsiUi(0) = 0 , dsi |D(0)| = 0 , ∀ i ∈ B− , (9e)
|D(Si)|dsiUi(Si) = 0 , dsi |D(Si)| = 0 , ∀ i ∈ B+ . (9f)
For the definition of the variational formulation of problem (9) we introduce Sobolev spaces defined ona metric graph, see for example [2, 43] and references therein. In particular H1(Λ) =
⊕ΛiH1(Λi) ∩ C0(Λ),
namely the space of all continuous functions V : Λ → R, such that their restriction Vi to each edgeΛi, i = 1, . . . , N belongs to H1(Λi), where H1(Λi) denotes the usual H1 space defined on the manifold Λi.
The norm of H1(Λ) is naturally defined as, ‖V ‖2H1(Λ) =∑Ni=1 ‖Vi‖2H1(Λi)
.
Let us now take U, V ∈ H1(Λ) and derive the variational formulation of (9). From (9a), we obtain,
N∑i=1
[(dsiUi, dsVi)Λi,|D| − dsi(|D(si)|Ui)Vi|si=Sisi=0 + dsi(|D(si)|)UiVi|si=Sisi=0 + (κiUi, Vi)Λi,|∂D|
]=
N∑i=1
(κiu⊕, Vi)Λi,|∂D| + (g, V )Λ,|D| .
After using (9b) and reordering the terms at the endpoints of each edge we have,
N∑i=1
[|D(Si)|dsiUi(Si)Vi(Si)− |D(0)|dsiUi(0)Vi(0)
]=
M∑j=1
[ ∑i∈K+
j
|D(Si)|dsiUi(Si)Vi(Si)−∑i∈K−j
|D(0)|dsiUi(0)Vi(0)
]
+∑i∈B+
|D(Si)|dsiUi(Si)Vi(Si)−∑i∈B−
|D(0)|dsiUi(0)Vi(0) .
Since the test functions are continuous on Λ, they can be factorized and all the terms on Kj and B disappearowing to (9c)-(9e)-(9f). Finally, conditions (9d) are stongly enforced through the definition of H1(Λ). Asa result of that, the variational formulation of the reduced problem on the network consists of findingU ∈ H1(Λ) such that
N∑i=1
[(dsiUi, dsVi)Λi,|D| + (κiUi, Vi)Λi,|∂D|
]=
N∑i=1
(κiu⊕, Vi)Λi,|∂D| + (g, V )Λ,|D| . (10)
Endowed with problem (10) the metric graph Λ becomes a quantum graph, namely a metric graph equippedwith a differential operator on the edges complemented with vertex conditions, see for example [3]. In thesimple case of constant cross sections, namely ds|∂D(s)| = 0, the differential operator on Λ is L(U) =|D|d2
ssU + |∂D|κU (also called as a Schrodinger-type or Hamiltonian operator) and the vertex conditions arethe Kirchhoff equations reported above.
8
3. Mathematical analysis and numerical approximation of the problem
We study the existence, uniqueness, stability and regularity of solutions of problem (6)-(8). The mainfocus of this section is to show how the formal derivation of the 3D-1D coupled problem actually provides amathematically sound definition of the coupling operators from a 3D domain, Ω, to a 1D manifold, Λ, andvice versa. This is a non trivial result, because the standard trace operator form a domain Ω to a subset Λis not well posed if Λ is a manifold of co-dimension two of Ω.
3.1. Coupled problems with hybrid dimensionality
Let us now introduce the following bilinear forms:
aΩ(w, v) = (∇w,∇v)Ω,
aΛ(w, v) = (dsw, dsv)Λ,|D|,
bεΛ(w, v) = (κw, v)Λ,|∂D|.
After averaging of the equation on Ω and of the interface conditions, for any f ∈ L2(Ω), g ∈ L2(Λ), theweak formulation of the reduced problem consists to find u ∈ H1
0 (Ω), U ∈ H1(Λ) such that
aΩ(u, v) + bεΛ(u, v) = bεΛ(U, v) + (f, v)Ω ∀v ∈ H10 (Ω) , (11a)
aΛ(U, V ) + bεΛ(U, V ) = bεΛ(u, V ) + (g, V )Λ,|D| ∀V ∈ H1(Λ) . (11b)
This problem is an extension to 3D of the one considered in [26] for two space dimensions.For what follows, it is convenient to introduce a compact formulation for problem (11). In particular, we
define V = [v, V ] a generic function of the space V = H10 (Ω)×H1(Λ) and we name U = [u, U ] the couple of
unknowns of problem (11). Any function V ∈ V is endowed with the norm |||V|||2 = ‖v‖2H1(Ω) + ‖V ‖2H1(Λ),|D|.
Then, we introduce the following bilinear form in V× V,
A(U ,V) = aΩ(u, v) + aΛ(U, V ) + bεΛ(u− U, v − V ) ,
and the linear functional in V, F(V) = (f, v)Ω +(g, V )Λ,|D|. Then, the compact form of problem (11) consistsof finding U ∈ V such that
A(U ,V) = F(V), ∀V ∈ V . (12)
3.2. Well-posedness analysis
The mathematical properties of problem (12) are studied below. Before proceeding, we list the mainassumptions on the domain and on the coefficients, at the basis of the existence of a solution (E1-E3). Fromnow on, the dependence of D and ∂D from the arclength s will be omitted in the notation.
E1: Λ ⊂⊂ Ω are bounded and domains in Rd−2 and Rd respectively, with d = 3.E2: The domain Σ is a generalized cylinder with strictly positive minimal diameter mins |D(s)| > 0 for any
s ∈ (0, S). As a consequence Σ has planar faces |D(0)|, |D(S)| > 0 at the endpoints of Λ.E3: There exist positive constants CD, C∂D independent of s, such that
|D| = CD (diam(D))2, |∂D| = C∂D diam(D) . (13)
E4: The parameter κ ∈ L∞(Λ) is strictly positive and lower bounded by κmin > 0.
Remark 3.1. Assumption E2 prevents that the generalized cylinder Σ pinches at any interior or extremallocation of the centerline Λ, i.e. the cross section of the cylinder can not collapse. In particular, this entailsthat the cylinder must have blunt (planar) tips. This is an essential requirement in the analysis, becausethe boundedness of the bilinear form depends of the minimal radius of the cylinder, as shown in Lemma3.7. This is due to the fact that, if a cross section collapses, then the coupling between the 3D and the 1Dproblem becomes ill posed. This is equivalent to observe that, if |D| → 0, then we loose control of U in theweighted norm |||U|||.
9
Remark 3.2. Assumption E3, namely that CD and C∂D are independent of s means that the cross sectionof Σ can not change its shape, but it can be subject to a homothetic map and rotation.
Theorem 3.3. Under the assumptions E1-E4, problem (12) has a unique solution U ∈ V satisfying thefollowing stability estimate,
|||U||| ≤ CS1
(‖f‖L2(Ω) + ‖g‖L2(Λ),|D|
)(14)
with
CS1 =2√
1 + (1 + CP (Ω))2C2T ‖κ‖2L∞
min(CT ‖κ‖L∞ , 2(1 + (1 + CP (Ω)) CT ‖κ‖L∞), C∂D/CDκmin
) ,where the meaning of constants will be clarified in what follows.
Before addressing the central result, we present some auxiliary tools that will be useful in the analysis.
Lemma 3.4. If v ∈ H1(Ω) or alternatively v ∈ L2(Γ), then v ∈ L2(Λ) and the following inequality holds
‖v‖2L2(Λ),|∂D| ≤ ‖v‖2L2(Γ) ≤ CT (Γ,Ω)‖v‖2H1(Ω), (15)
being CT (Γ,Ω) the (positive) constant of the trace inequality from L2(Γ) to H1(Ω). Moreover, CT (Γ,Ω) =
O(ε1−1
2−δ ), for δ > 0 sufficiently small, therefore it tends to 0 when ε tends to 0.
Proof. If the inequality (15) holds, it follows immediately that v ∈ L2(Λ), since v ∈ H1(Ω), or alternativelyv ∈ L2(Γ). Therefore, we consider
‖v‖2L2(Λ),|∂D| =
∫Λ
|∂D|v2 ds =
∫Λ
1
|∂D|
(∫∂D
v dγ
)2
ds. (16)
Using Jensen’s inequality, we obtain∫Λ
1
|∂D|
(∫∂D
v dγ
)2
ds ≤∫
Λ
∫∂D
v2 dγ ds (17)
and consequently
‖v‖2L2(Λ),|∂D| ≤∫
Λ
∫∂D
v2 dγ ds = ‖v‖2L2(Γ) ≤ CT (Γ,Ω)‖v‖2H1(Ω). (18)
We now prove that the constant CT (Γ,Ω) tends to 0 when ε tends to 0 analyzing the behaviour of the traceconstrant C(Γ,Ω⊕) from L2(Γ) to H1(Ω⊕) and exploiting the fact that for any function v ∈ H1(Ω) we have
‖v‖2L2(Γ) ≤ CT (Γ,Ω⊕)‖v‖2H1(Ω⊕) ≤ CT (Γ,Ω⊕)‖v‖2H1(Ω) .
Therefore, let v ∈ H1(Ω⊕), then the trace of v on Γ is in H12 (Γ) and by Sobolev embedding theorem we
have that H12 (Γ) is embedded in Lp(Γ) for p ≤ 4. Using Holder inequality with positive exponents 2− δ and
2−δ1−δ , for δ > 0 sufficiently small,
‖v‖2L2(Γ) =
∫Γ
v2d σ ≤(∫
Γ
12−δ1−δ dσ
)1− 12−δ(∫
Γ
v4−2δ
) 12−δ
≤ |Γ|1−1
2−δ ‖v‖2L4−2δ(Γ)
≤ |Γ|1−1
2−δCT (∂Ω⊕,Ω⊕)‖v‖2H1(Ω⊕),
where CT (∂Ω⊕,Ω⊕) is bounded as ε → 0. Indeed, the problem of studying the asymptotic behaviourof the constant in the trace inequality from L4−2δ(∂Ω⊕) to H1(Ω⊕) as ε → 0 can be reformulated as theproblem of studying the behaviour of the trace constant as the external boundary ∂Ω expands (we recall that∂Ω = ∂Ω⊕ \ Γ). Then, from [16, Theorem 1.3, item 2], taking p = 2, q = 4− 2δ, we have that CT (∂Ω⊕,Ω⊕)is uniformly bounded with respect to the size of the domain and the relative size of the boundary. Therefore
CT (Γ,Ω⊕) ≤ |Γ|1−1
2−δCT (∂Ω⊕,Ω⊕) = O(ε1−1
2−δ ) when ε→ 0, because the surface area of Γ is proportionalto ε.
10
Lemma 3.5 (Poincare inequality). For any v ∈ H10 (Ω), there exists a positive constant, CP (Ω), s.t.
‖v‖2L2(Ω) ≤ CP (Ω)‖∇v‖2L2(Ω).
We now address the well-posedness of problem (12) on the basis of the theory for linear variationalproblems in Banach spaces. More precisely we use Theorem 2.6 of [15] (also named the Banach-Necas-Babuska Theorem), which for the sake of clarity is adapted here to the notation used for (12).
Theorem 3.6. Let V be a reflexive Banach space; let A(·, ·) be a bounded bilinear form on V× V; let F bea bounded linear functional on V, i.e. F ∈ V′. Problem (12) is well-posed if and only if:
∃α > 0 : infW∈V
supV∈V
A(W,V)
|||W||| |||V|||≥ α , (BNB1)
∀V ∈ V :(A(W,V) = 0 ∀W ∈ V
)⇒ V = 0 . (BNB2)
We first analyze the boundedness of the bilinear form and of the functional.
Lemma 3.7. Under the assumptions E1-E4, the bilinear form A is bounded with respect to the norm |||·|||,
A(U ,V) ≤ |||A||| |||U||| |||V||| , ∀U ,V ∈ V ,
by the constant
|||A||| = 2 + ‖κ‖L∞(√
CT (Γ,Ω) +
√C∂D
CDmins (diam(D))
)2
.
The functional F is bounded in V owing to the inequality,
F(V) ≤(‖f‖L2(Ω) + ‖g‖L2(Λ),|D|
)|||V|||.
Proof. We start by recalling the definition of the global bilinear form,
A(U ,V) = aΩ(u, v) + aΛ(U, V ) + bεΛ(u− U, v − V ).
For the first and the second terms it easily follows that
aΩ(u, v) ≤ |||U||||||V||| , aΛ(U, V ) ≤ |||U||||||V||| .
For the last term we have
bεΛ(u− U, v − V ) = (κ(u− U), v − V )Λ,|∂D| ≤ ‖κ‖L∞‖u− U‖L2(Λ),|∂D|‖v − V ‖L2(Λ),|∂D|
≤ ‖κ‖L∞(‖u‖L2(Λ),|∂D| + ‖U‖L2(Λ),|∂D|
) (‖v‖L2(Λ),|∂D| + ‖V ‖L2(Λ),|∂D|
).
We notice that, using (13), ∀V ∈ H1(Λ)
‖V ‖2L2(Λ),|∂D| =
∫Λ
|∂D||D||D|V 2 ds =
C∂DCD
∫Λ
1
diam(D)|D|V 2 ds
≤ C∂DCDmins (diam(D))
‖V ‖2L2(Λ),|D| ≤C∂D
CDmins (diam(D))‖V ‖2H1(Λ),|D|.
(19)
Consequently, using Lemma 3.4,
‖v‖L2(Λ),|∂D| + ‖V ‖L2(Λ),|∂D| ≤√CT (Γ,Ω)‖v‖H1(Ω) +
√C∂D
CDmins (diam(D))‖V ‖H1(Λ),|D|
≤
(√CT (Γ,Ω) +
√C∂D
CDmins (diam(D))
)|||V||| .
(20)
11
Therefore we have,
bεΛ(u− U, v − V ) ≤ ‖κ‖L∞(√
CT (Γ,Ω) +
√C∂D
CDmins (diam(D))
)2
|||U||||||V|||.
The boundedness of F is easily shown owing to Cauchy-Schwarz inequality,
F(V) = (f, v)Ω + (g, V )Λ,|D| ≤ ‖f‖L2(Ω)‖v‖L2(Ω) +‖g‖L2(Λ),|D|‖V ‖L2(Λ),|D| ≤(‖f‖L2(Ω) + ‖g‖L2(Λ),|D|
)|||V|||.
Lemma 3.8. Under the assumptions E1-E4, the operator A satisfies (BNB1)-(BNB2) with a constant αindependent of ε.
Proof. In order to prove that the bilinear form A satisfies (BNB1) it is sufficient to prove that there existsa positive constant α such that ∀W ∈ V we can find V ∈ V satisfying
A(W,V)
|||W||| |||V|||≥ α.
We subdivide the proof in the following steps. We prove that:
(i) ∃m1, m2, m3 > 0 :
A(V,V) ≥ m1‖v‖2H1(Ω) +m2|V |2H1(Λ),|D| +m3‖v − V ‖2L2(Λ),|∂D| , ∀V ∈ V. (21)
(ii) ∀W ∈ V, ∃V ∈ V and α1 > 0 :
A(W,V) ≥ α1|||W|||2 (22)
(iii) and ∃α2 > 0 :
|||W||| ≥ α2|||V|||.
From the last two inequalities we obtain that (BNB1) holds for α = α1α2. In details:(i) By definition of A,
A(V,V) = aΩ(v, v) + aΛ(V, V ) + bεΛ(v − V, v − V )
and for the first term we have
aΩ(v, v) = (∇v,∇v)Ω ≥ (1 + CP (Ω))−1‖v‖2H1(Ω),
where CP (Ω) is the Poincare constant. For the second term we have,
aΛ(V, V ) = |V |2H1(Λ),|D| . (23)
Finally, for the last one we have
bεΛ(v − V, v − V ) = (κ(v − V ), v − V )Λ,|∂D| ≥ κmin‖v − V ‖2L2(Λ),∂D.
Therefore (21) holds and m1 = (1 + CP (Ω))−1, m2 = 1, m3 = κmin.12
(ii) For any W = [w,W ], we choose V =W + δ[0,W ] and from (i) we have
A(W,W + δ[0,W ]) =A(W,W) + δA(W, [0,W ])
≥m1‖w‖2H1(Ω) +m2|W |2H1(Λ),|D| +m3‖w −W‖2L2(Λ),|∂D|
+ δ (aΩ(w, 0) + aΛ(W,W ) + bεΛ(w −W,−W ))
≥m1‖w‖2H1(Ω) +m2|W |2H1(Λ),|D|
+ δ(|W |2H1(Λ),|D| + (κ(w −W ),−W )Λ,|∂D|
)≥m1‖w‖2H1(Ω) + (m2 + δ)|W |2H1(Λ),|D|
− δ((κw,W )Λ,|∂D| − (κW,W )Λ,|∂D|
).
(24)
We notice that using Young inequality and Lemma 3.4 we obtain
(κw,W )Λ,|∂D| =
∫Λ
|∂D|κwW ds ≤ 1
2
(∫Λ
|∂D|κw2 ds+
∫Λ
|∂D|κW 2 ds
)≤1
2
(‖κ‖L∞‖w‖2L2(Λ),|∂D| + (κW,W )Λ,|∂D|
)≤1
2
(CT (Γ,Ω)‖κ‖L∞‖w‖2H1(Ω) + (κW,W )Λ,|∂D|
).
As a consequence of Lemma (3.4), there exists an upper bound CT for the trace constant CT (Γ,Ω) indepen-dent of ε. Therefore, we have
(κw,W )Λ,|∂D| ≤1
2
(CT ‖κ‖L∞‖w‖2H1(Ω) + (κW,W )Λ,|∂D|
).
Substituting in (24) we have,
A(W,W + δ[0,W ]) ≥(m1 −
δ
2CT ‖κ‖L∞
)‖w‖2H1(Ω) + (m2 + δ)|W |2H1(Λ),|D| +
δ
2(κW,W )Λ,|∂D|
and we choose δ = m1
CT ‖κ‖L∞so that m1 − δ
2 CT ‖κ‖L∞ = m1
2 and it is positive. Using assumption (13) and
reminding that diam(D) ≤ 1, we have,
(κW,W )Λ,|∂D| ≥ κmin∫
Λ
|∂D|W 2 ds = κmin
∫Λ
C∂DCD diam(D)
|D|W 2 ds ≥ C∂DCD
κmin‖W‖2L2(Λ),|D|. (25)
Therefore
A(W,W + δ[0,W ]) ≥m1
2‖w‖2H1(Ω)
+ (m2 + δ)|W |2H1(Λ),|D| +δ
2
C∂DCD
κmin‖W‖2L2(Λ),|D|
≥m1
2‖w‖2H1(Ω) + min
(m2 + δ,
δ
2
C∂DCD
κmin
)‖W‖2H1(Λ),|D|
≥α1|||W|||2
with α1 = min(m1
2 , m2 + δ, δ2C∂DCD
κmin
)= min
(1
2(1+CP (Ω)) , 1 + 1(1+CP (Ω))CT ‖κ‖L∞
, 12
C∂DκminCD(1+CP (Ω))CT ‖κ‖L∞
).
(iii) We show that there exists a constant α2 such that
|||W||| ≥ α2|||W + δ[0,W ]|||. (26)13
The inequality above can be proved as follows:
|||W + δ[0,W ]|||2 ≤ |||W|||2 + δ2|||[0,W ]|||2 = ‖w‖2H1(Ω) + (1 + δ2)‖W‖2H1(Λ),|D| ≤ (1 + δ2)|||W|||2.
Therefore (26) holds with α2 = (√
1 + δ2)−1 =
(√1 +
(1
(1+CP (Ω))CT ‖κ‖L∞
)2)−1
. Being α1 and α2 inde-
pendent of ε, it follows that also α is independent of ε.For the proof of (BNB2) we choose W = V and being A(V,V) = 0, from (i) we have
m1‖v‖2H1(Ω) +m2|V |2H1(Λ),|D| +m3‖v − V ‖2L2(Λ),|∂D| = 0,
and consequently
‖v‖H1(Ω) = 0, |V |H1(Λ),|D| = 0, ‖v − V ‖L2(Λ),|∂D| = 0. (27)
Then, v = 0 and |V |H1(Λ),|D| = 0 with ‖V ‖2L2(Λ),|∂D| = 0 imply V = 0.
Combining Lemma 3.8 and Theorem 3.6, we obtain the well-posedness of (12). In order complete theproof of Theorem 3.3, it remains to show that the stability estimate (14) holds. Hence,
|||U||| ≤ 1
αsupV∈V
A(U ,V)
|||V|||=
1
αsupV∈V
F(V)
|||V|||≤ 1
α
(‖f‖L2(Ω) + ‖g‖L2(Λ),|D|
)where the last inequality follows from Lemma 3.7.
3.3. Additional regularity of the solution of the problem in Ω
We observe that the weak formulation (11a) could have been formally written in strong form as
−∆u = f − κEΓ (u− U) δΓ in Ω, u = 0 on ∂Ω, (28)
where δΓ is the Dirac measure of the surface Γ and EΓ denotes an extension operator from Λ to Γ. Moreprecisely, given a continuous function ϕ ∈ C0(Λ), for any s ∈ (0, S) the extension operator is such that
EΓϕ(r, t; s) = ϕ(s) ∀r ∈ (0, R), t ∈ (0, T ),
namely, the extension operator spans the point-wise value ϕ(s) on λ(s) + ∂D(s), preserving the regularityof the function. It is straightforward to show that (κEΓ (u− U) δΓ, v)Ω becomes bεΛ(u−U, v) in the variationalformulation, as follows∫
Ω
κEΓ (u− U) vδΓ dΩ =
∫Γ
κEΓ(u− U)v dσ =
∫Λ
κ(u− U)
∫∂D(s)
v dγ ds =
∫Λ
|∂D|κ(u− U)v ds.
Due to the presence of the Dirac source δΓ, global H2-regularity can not be recovered and the questionarises to which interspace X with H2(Ω) ⊂ X ⊂ H1
0 (Ω) the solution u belongs to. This property will beaddressed in Theorem 3.11. Our analysis is based on the following regularity properties of elliptic equationsstudied in [22]. More precisely, we introduce two auxiliary Lemmas, which in turn require some additionalassumption on the regularity of domains, listed below.
Lemma 3.9. Let Ω be a generic bounded, convex domain in Rd. Let γ ⊂ Ω be a C2-surface such that thedistance between γ and ∂Ω is positive and γ ⊂ ∂D for some three-dimensional C2-domain D ⊂⊂ Ω. Considerthe following problem
−∆y = φδγ in Ω
y = 0 on ∂Ω,(29)
14
with δγ being the Dirac measure of γ and φ(x) ∈ L2(γ). Problem (29) has a unique solution y and y ∈H
32−η(Ω) ∩H1
0 (Ω) for each η > 0. Furthermore the function y, suitably extended to Rd, can be decomposedas
y = z + w with z ∈ H2(Ω), w ∈ H 32−η(Rd)
respectively solutions of −∆z = −y∆(1− F )− 2∇(1− F )∇y in Ω,
z = 0 on ∂Ω,
−∆w = −y∆F − 2∇F∇y + φδγ in Rd,
where F is a regular function such that −y∆(1−F )−2∇(1−F )∇y ∈ L2(Ω) and −y∆F −2∇F∇y ∈ L2(Rd).
Proof. For the proof see [22, Theorem 2.1, case (iii)].
Lemma 3.10. Under the assumptions of Lemma 3.9 and the additional assumption that Ω has a C2 boundary∂Ω, problem (29) satisfies the following stability estimate in H
32−η(Ω): there exists a positive constant CR
such that‖y‖
H32−η(Ω)
≤ CR‖φδγ‖H−
12−η(Ω)
. (30)
Proof. From the decomposition of Lemma 3.9 we have
‖y‖H
32−η(Ω)
≤ ‖z‖H2(Ω) + ‖w‖H
32−η(Rd)
.
For the sake of simplicity, we will denote the positive constants appearing in the following inequalitiesalways with the simbol CR, even if they might have different meanings and values. Being ∂Ω of class C2,since z ∈ H1
0 (Ω), from [20, Theorem 8.12] we obtain that there exists a constant CR such that
‖z‖H2(Ω) ≤ CR(‖z‖L2(Ω) + ‖y∆(1− F ) + 2∇(1− F )∇y‖L2(Ω)
).
Then, owing to the standard H1-stability of the Poisson problem with right-hand side in L2(Ω), we obtain
‖z‖H2(Ω) ≤ CR‖y∆(1− F ) + 2∇(1− F )∇y‖L2(Ω) ≤ CR‖y‖H1(Ω).
Still exploiting the H1-stability of the Poisson problem with H−1(Ω) right-hand side, reminding that φδγ ∈H−
12−η(Ω) ⊂ H−1(Ω), we obtain
‖y‖H1(Ω) ≤ CR‖φδγ‖H− 12−η(Rd)
and consequently‖z‖H2(Ω) ≤ CR‖φδγ‖H− 1
2−η(Rd)
.
Concerning the upper bound of the function w, we exploit the theory of pseudo-differential operators. Moreprecisely, we use the generalization of the Garding inequality to fractional Sobolev spaces, see in particularProposition 5.4 of [1] and its consequences, to show that there exists a constant CR such that
‖w‖H
32−η(Rd)
≤ CR(‖∆w‖
H−12−η(Rd)
+ ‖w‖H1(Rd)
).
Then we have
‖∆w‖H−
12−η(Rd)
≤ ‖y∆F + 2∇F∇y‖L2(Ω) + ‖φδγ‖H−
12−η(Rd)
≤ CR‖y‖H1(Ω) + ‖φδγ‖H−
12−η(Rd)
and for the stability of the Poisson problem in H1(Rd) we obtain
‖w‖H1(Rd) ≤ CR(‖y‖H1(Ω) + ‖φδγ‖
H−12−η(Rd)
).
15
Therefore,
‖w‖H
32−η(Rd)
≤ CR(‖y‖H1(Ω) + ‖φδγ‖
H−12−η(Rd)
)≤ CR‖φδγ‖
H−12−η(Rd)
Putting together all the previous estimates, we obtain (30).
In order to apply Lemma 3.9 to our case, we require that the domains Ω and Σ satisfy the followingregularity assumptions (R1-R3):
R1: Ω is a convex domain in Rd with C2-regular boundary ∂Ω.R2: Σ ∈ Rd has a C2-regular boundary ∂Σ.R3: Σ is completely embedded into Ω, such that the distance between ∂Ω and ∂Σ is strictly positive.
Theorem 3.11. Under the assumptions R1-R3, in addition to those of Theorem 3.3, the sub-problem on Ωenjoys additional regularity u ∈ H 3
2−η(Ω) for any η > 0 and the following estimate is satisfied,
‖u‖H
32−η(Ω)
≤ CS2
(‖f‖L2(Ω) + ‖g‖L2(Λ),|D|
)(31)
where
CS2 := CR
1 + 2 max
(√CT (Γ,Ω)C ′T (Γ,Ω),
√C ′T (Γ,Ω)C∂D
CD mins (diam(D))
)CS1
.
Proof. The proof of Theorem 3.11 is a direct consequence of Lemmas 3.9 and 3.10. Owing to Theorem 3.3we have −κEΓ (u− U) ∈ L2(Γ), so that Lemma 3.9 can be directly applied to equation (28). For the upperbound, inequality (30) implies that
‖u‖H
32−η(Ω)
≤ CR(‖f‖L2(Ω) + ‖EΓuδΓ‖
H−12−η(Ω)
+ ‖EΓUδΓ‖H−
12−η(Ω)
). (32)
From the definition of ‖ · ‖H−
12−η(Ω)
, we have
‖EΓuδΓ‖H−
12−η(Ω)
= supw∈H
12
+η(Ω)w 6=0
(EΓu,w)Γ
‖w‖H
12
+η(Ω)
.
Then, we apply trace inequalities to derive the following upper bound,
(EΓu,w)Γ ≤ ‖u‖L2(Λ),|∂D|‖w‖L2(Γ) ≤√CT (Γ,Ω)C ′T (Γ,Ω)‖u‖H1(Ω)‖w‖H 1
2+η(Ω)
,
where C ′T (Γ,Ω) is the constant in the trace inequality from L2(Γ) to H12 +η(Ω). Reasoning as in Lemma 3.4
for CT (Γ,Ω), we observe that C ′T (Γ,Ω) is (at least) upper bounded when ε tends to 0. Proceeding similarlyfor the last term of (32) and using (19) we obtain,
‖EΓUδΓ‖H−
12−η(Ω)
≤√C ′T (Γ,Ω)‖U‖L2(Λ),|∂D| ≤
√C ′T (Γ,Ω)C∂D
CD mins (diam(D))‖U‖H1(Λ),|D|
Putting together the previous inequalities we obtain,
‖u‖H
32−η(Ω)
≤ CR
‖f‖L2(Ω) + 2 max
[√CT (Γ,Ω)C ′T (Γ,Ω),
√C ′T (Γ,Ω)C∂D
CD mins (diam(D))
]|||U|||
,
that combined with (14) gives the desired result.
16
4. Model error analysis
We study the error introduced by replacing the model (1) with (11). We recall the a posteriori analysisof modeling error developed in [7] for general abstract problems and we apply it here to the particular caseof topological model reduction for small cylindrical inclusions. The main objective is to characterize thedependence of the modeling error on the radius of the inclusion, namely ε. To this purpose, we split themodeling error in three components, corresponding the assumptions A1, A2, A3 described in Section 2. Eachcomponent will be analyzed in one of the following subsections.
Before proceeding, we recall the abstract theory developed in [7], for the particular case of linear ellipticproblems. Let u be the solution of the reduced problem and let uref be the one of the reference problem,which are respectively defined as follows:
find uref ∈ X : aref(uref , v) = Fref(v), ∀v ∈ X, (33)
find u ∈ X : a(u, v) = F(v), ∀v ∈ X, (34)
where X is a suitable Hilbert space, X ′ its dual space and aref and a are the bilinear forms that characterizethe reference and reduced problem respectively. We assume that aref and Fref can be expressed as amodification of the form a and the functional F as follows
aref(u, v) = a(u, v) + d(u, v), ∀u, v ∈ X ,
Fref(v) = F(v) + l(v), ∀v ∈ X ,
As a result, the modeling error e = uref − u ∈ X satisfies the following equation,
a(e, v) + d(uref , v) = l(v), ∀v ∈ X .
Let j(·) : X → R be a linear functional. We aim to estimate the modeling error measured by j(e) using thedual weighted residual approach. We define the dual problem with respect to (33),
find zref ∈ X : aref(v, zref) = j(v), ∀v ∈ X .
Following the approach presented in [7], under the assumptions stated above, we obtain that the error outputfunctional is represented as follows,
j(e) = aref(e, zref) = a(e, zref) + d(uref − u, zref) = l(zref)− d(u, zref) . (35)
The aim of this section is to analytically characterize the asymptotic behavior of the error when the smallparameter ε → 0+. We introduce two fundamental properties that will be analyzed case by case later on.One is that the bilinear form d and the functional l are bounded in suitable norms ‖·‖? and ‖·‖. The secondproperty concerns stability estimates for the solutions of the reduced primal and reference dual problems.
Property 4.1. There exist constants ‖d‖?, ‖l‖ such that
d(u, zref) ≤ ‖d‖?‖u‖? ‖zref‖, l(zref) ≤ ‖l‖‖zref‖ . (36)
Furthermore, the constants ‖d‖? and ‖l‖ asymptotically vanish with ε→ 0.
Property 4.2. There exist constants ‖a‖? and ‖aref‖, uniformly bounded with ε, such that
‖u‖? ≤ ‖a‖?‖F‖X′ , ‖zref‖ ≤ ‖aref‖‖j‖X′ . (37)
Combining Properties 4.1 and 4.2 we obtain the characterization of the asymptotic behavior of the mod-eling error with respect to ε. We are particularly interested in the case j(v) = (e, v)L2 , because it allows usto derive the following bound,
‖e‖2L2 = j(e) = l(zref)−d(u, zref) ≤ ‖l‖‖zref‖+‖d‖?‖u‖?‖zref‖ ≤ (‖l‖ + ‖d‖?‖a‖?‖F‖X′) ‖aref‖‖e‖L2 ,17
which directly entails the L2 control of the modeling error.We will pursue this analysis for the three sources of error, corresponding to assumptions A1, A2, A3. As a
result, we will study three different operators d(k)(·, ·) and l(k)(·) with k = 1, 2, 3, each one corresponding toa different source of error. Before proceeding, we introduce some results that will be useful in what follows.
Lemma 4.1 (Stekloff inequality [29]). Let ∂D be an ellipse or a rhombus. There exists a positive constantC such that for any function v ∈ H1(D) satisfying∫
∂Dvdγ = 0 , (38)
the following inequality holds ∫∂D
v2dγ ≤ C∫D
(∇v)2dσ .
More precisely if ∂D is an ellipse (x1/a1)2 + (x2/a2)2 = 1 then C ≤ max[a1, a2]. If ∂D is a rhombus
|x1/a1|+ |x2/a2| = 1, then C ≤ (a21 + a2
2)12 /min[a1/a2, a2/a1].
From the previous lemma combined with assumption A0, we conclude that there exists CS , independentof ε, such that for any v ∈ Hk(Σ), with k ≥ 3
2 , satisfying (38), we have
∫Γ
v2dσ ≤ CSε∫
Σ
(∇v)2dω . (39)
Lemma 4.2 (Poincare-Wirtinger inequality [34]). Let D ∈ R2 be a convex domain of diameter D. For anyfunction v ∈ H1(D) such that ∫
Dvdσ = 0 ,
we have, ∫Dv2dσ ≤ D2
π2
∫D
(∇v)2dσ . (40)
Lemma 4.3 (Extension theorem for domains having small geometric details [41]). Let s be a nonnegativeinteger. There exists an extension operator EΣ from Hs(Ω⊕) to Hs(Σ) such that
‖EΣv‖Hs(Σ) ≤ CE‖v‖Hs(Ω⊕) ∀v ∈ Hs(Ω⊕)
where the constant CE is independent of Σ = Ω \ Ω⊕.
Lemma 4.4 (Poincare-Friedrichs inequality [8, 34]). There exists a positive constant CPF (Σ) such that forany v ∈ H1(Σ) it holds,
‖v‖2L2(Σ) ≤ CPF (Σ)(‖∇v‖2L2(Σ) + ‖v‖2L2(Γ)
)(41)
and CPF (Σ) tends to 0 for ε→ 0.
Proof. Integrating by parts we have:
‖v‖2L2(Σ) =1
2
2∑i=1
∫Λ
∫Dv2 · 1 dσ ds = −1
2
2∑i=1
∫Λ
∫D
2v∂v
∂xixi dσ ds+
1
2
2∑i=1
∫Λ
∫∂D
v2xini dγ ds,
18
where x1, x2 is a generic system of coordinates for D and ni is the i−th component of the outward unitsurface normal to D. Then, using Schwarz and Young inequalities we obtain
‖v‖2L2(Σ) ≤1
2
2∑i=1
2ε‖v‖L2(Σ)
∥∥∥∥ ∂v∂xi∥∥∥∥L2(Σ)
+1
2
2∑i=1
ε‖v‖2L2(Γ)
≤ 1
2‖v‖2L2(Σ) +
1
2
2∑i=1
2ε2∥∥∥∥ ∂v∂xi
∥∥∥∥2
L2(Σ)
+ ε‖v‖2L2(Γ)
≤ 1
2‖v‖2L2(Σ) + ε2‖∇v‖2L2(Σ) + ε‖v‖2L2(Γ)
from which it follows that
‖v‖2L2(Σ) ≤ max(2ε2, 2ε
) (‖∇v‖2L2(Σ) + ‖v‖2L2(Γ)
).
Thus, CPF (Σ) = max(2ε2, 2ε
)and it tends to 0 when ε tends to 0.
4.1. Analysis of the modeling error of the one dimensional problem (assumption A1)
We aim to study the modeling error of replacing the equation (1b) with (6). Since this analysis referto the assumption A1, we will endow the general operators of the previous section with the apex (1). Thereduced problem (reported here for the sake of clarity) is to find U ∈ H1(Λ) such that
(dsU, dsV )Λ,|D| + (κU, V )Λ,|∂D| = (κu⊕, V )Λ,|∂D| + (g, V )Λ,|D| ∀V ∈ H1(Λ) .
where u⊕ is the weak solution of (1) and it belongs to the space of functions of H1(Ω⊕) with null trace on∂Ω, denoted as H1
∂Ω(Ω⊕). The reference problem in the weak form consists of finding u ∈ H1(Σ) such that
(∇u,∇v)Σ + (κu, v)Γ = (κu⊕, v)Γ + (g, v)Σ, ∀v ∈ H1(Σ) .
We define the modeling error e(1) as the difference between u and U . To this purpose, we exploit thecylindrical configuration of the domain Σ and its local coordinate system. In particular, we uniformly extendU(s) on every cross section D(s) of the cylinder and with abuse of notation we still denote the extendedfunction with U . Thanks to the regularity of U on Λ, we have that the extension on Σ belongs to H1(Σ).Referring to the general notation introduced in the previous section we have that the solutions of the reference
and reduced models are u(1)ref = u and u(1) = U both in the space X(1) = H1(Σ). As a result, the modeling
error is e(1) = u(1)ref − u(1) = u − U ∈ H1(Σ). The bilinear forms of the reference and reduced problems are
a(1)ref (u, v) = (∇u,∇v)Σ + (κu, v)Γ ,
a(1)(u, v) = (dsu, ∂sv)Λ,|D| + (ν(u− u), ∂sv)Λ,|∂D| + (κu, v)Λ,|∂D| ,
= (dsu, ∂sv)Σ + (ν(u− u), ∂sv)Γ + (κu, v)Γ ,
where a(1)(u, v) is a generalization of the bilinear form of the reduced problem that can be applied to anyfunction u, v ∈ H1(Σ). More precisely, it is the generalization to any test function v ∈ H1(Σ) of (5).When applied to U, V ∈ H1(Λ) the bilinear form a(1)(U, V ) coincides with (6). We aim to quantify the
difference a(1)ref (u, v) − a(1)(u, v). Using the expression of the gradient in cylindrical coordinates ∇(·) =
[∂s(·), ∂r(·), r−1∂θ(·)]′, the difference operator d(1)(·, ·) becomes
d(1)(u, v) = a(1)ref (u, v)− a(1)(u, v)
= (∂su− dsu, ∂sv)Σ − (ν(u− u), ∂sv)Γ + (κ(u− u), v)Γ + (∂ru, ∂rv)Σ + (r−1∂θu, r−1∂θv)Σ .
(42)
19
Similarly, the difference l(1)(·) of the right hand sides is
l(1)(v) = ((I − (·))g, v)Σ + (κ(I − (·))u⊕, v)Γ . (43)
We now aim to prove properties 36 and 37 for the operators d(1), l(1) and a(1), a(1)ref respectively.
Lemma 4.5. The operator d(1)(·, ·) satisfies Property 4.1 with ‖d(1)‖? = 0.
Proof. Let U ∈ H1(Σ) be the extension to Σ of the reduced problem on Λ. We observe that d(U, v) = 0 for
any v ∈ H1(Σ), because ∂rU = ∂θU = 0 and U = U = U .
Lemma 4.6. Under the assumption that g ∈ H32 (Σ) and u⊕ ∈ H1
∂Ω(Ω⊕) ∩ H2(Ω⊕), the operator l(1)(·)satisfies Property 4.1 with the norm ‖ · ‖2 = ‖ · ‖2L2(Σ) + ‖ · ‖2L2(Γ) and with the constant
‖l(1)‖ =ε
π‖g‖H1(Σ) + ‖κ‖L∞CE
√CSε‖u⊕‖H1(Ω⊕) .
Proof. For the upper bound of the right hand side, we observe that g ∈ H 32 (Σ) implies that g|D ∈ H1(D).
Then, (I − (·))g satisfies the assumptions of the Poincare-Wirtinger inequality, which allows us to concludethat
((I − (·))g, v)Σ ≤(∫
Σ
((I − (·))g)2) 1
2(∫
Σ
v2) 1
2
=(∫
Λ
∫D
((I − (·))g)2dσds) 1
2 ‖v‖L2(Σ)
≤(∫
Λ
(diam(D))2
π2
∫D
(∇Dg)2dσds) 1
2 ‖v‖L2(Σ) ≤ε
π‖g‖H1(Σ)‖v‖ ,
where, from now on ∇D(·) = [∂r(·), r−1∂θ(·)]′ denotes the gradient in a local coordinate system of D.For the second term of l(1) we use the Stekloff inequality,
(κ(I − (·))u⊕, v)Γ ≤ ‖κ‖L∞(∫
Γ
(I − (·))u⊕)2) 1
2(∫
Γ
v2) 1
2
≤ ‖κ‖L∞√CSε
(∫Λ
∫D
(∇DEΣu⊕)2dσds) 1
2 ‖v‖L2(Γ)
≤ ‖κ‖L∞√CSε‖EΣu⊕‖H1(Σ)‖v‖L2(Γ) ≤ ‖κ‖L∞CE
√CSε‖u⊕‖H1(Ω⊕)‖v‖ ,
where EΣ denotes the extension operator of Lemma 4.3. The previous inequalities show that (36) is verifiedfor this component of the modeling error with the constant,
‖l(1)‖ =ε
π‖g‖H1(Σ) + ‖κ‖L∞CE
√CSε‖u⊕‖H1(Ω⊕) .
Lemma 4.7. The dual problem, that is to find z(1)ref ∈ H1(Σ) such that
(∇v,∇z(1)ref )Σ + (κv, z
(1)ref )Γ = j(1)(v), ∀v ∈ H1(Σ) , (44)
with j(1)(v) = (e, v)Σ, satisfies the following stability estimate with the norm ‖ · ‖ = ‖ · ‖L2(Σ) + ‖ · ‖L2(Γ)
‖z(1)ref ‖ ≤
2
min(2C−1
PF (Σ), C−1PF (Σ)κmin, κmin
)‖e‖L2(Σ) .
As a result, Property 4.2 is satisfied with the following constant, which is bounded for ε→ 0,
‖a(1)ref ‖ =
2
min(2C−1
PF (Σ), C−1PF (Σ)κmin, κmin
) .20
Proof. Owing to Poincare -Friedrichs inequality (41) we have,
aref(v, v) = (∇v,∇v)Σ + (κv, v)Γ ≥ min
(1,
1
2κmin
)C−1PF (Σ)‖v‖2L2(Σ) +
1
2κmin‖v‖2L2(Γ),
≥ 1
2min
(2C−1
PF (Σ), C−1PF (Σ)κmin, κmin
)‖v‖2, ∀v ∈ H1(Σ) .
As a result of that we obtain,
1
2min
(2C−1
PF (Σ), C−1PF (Σ)κmin, κmin
)‖z(1)
ref ‖2 ≤ aref(z
(1)ref , z
(1)ref ) = (e, z
(1)ref )Σ
≤ ‖e‖L2(Σ)‖z(1)ref ‖L2(Σ) ≤ ‖e‖L2(Σ)‖z
(1)ref ‖ ,
from which it follows that
‖a(1)ref ‖ =
2
min(2C−1
PF (Σ), C−1PF (Σ)κmin, κmin
)and for Lemma 4.4 the constant ‖a(1)
ref ‖ is bounded for ε→ 0.
4.2. Analysis of the modeling error relative to the domain (assumption A2)
This component of the modeling error arises because we identify the domain Ω⊕ of (1) with the entiredomain Ω. Let us assume for simplicity that f ∈ H1(Ω⊕) and let us denote by EΣf its extension to H1(Σ).In this case the reference and reduced models are respectively
find u⊕ ∈ H1∂Ω(Ω⊕) : (∇u⊕,∇v)Ω⊕ + (κu⊕, v)Γ = (f, v)Ω⊕ + (κU, v)Γ, ∀v ∈ H1
∂Ω(Ω⊕), (45)
find u(2) ∈ H10 (Ω) : (∇u(2),∇v)Ω + (κu(2), v)Γ = ((IΩ⊕ + EΣ)f, v)Ω + (κU, v)Γ, ∀v ∈ H1
0 (Ω) . (46)
We extend the solution of (45) from Ω⊕ to Ω, with the extension operator EΣ which takes the function u⊕on Ω⊕ and extends it to the interior of the domain. Then, the reference solution relative to assumption A2 is
u(2)ref = (IΩ⊕ +EΣ)u⊕ ∈ H1
0 (Ω). The solution of the reduced problem is instead u(2) ∈ H10 (Ω). The functional
space where we set the modeling error is X(2) = H10 (Ω) and error is e(2) = u
(2)ref −u(2) = (IΩ⊕+EΣ)u⊕−u(2) ∈
H10 (Ω). Then, subtracting (46) from (45), it is straightforward to determine the expression of the difference
operators d(2)(u, v) and l(2)(v),
d(2)(u, v) = −(∇u,∇v)Σ , l(2)(v) = −(EΣf, v)Σ ∀u, v ∈ H10 (Ω) . (47)
Lemma 4.8. The operators d(2)(·, ·) and l(2)(·) satisfy Property 4.1 with the norms ‖ · ‖? = ‖ · ‖H
32−η(Ω)
,
fora any η > 0, and ‖ · ‖ = ‖ · ‖H1(Ω⊕) and inequality (36) is satisfied with constants
‖d(2)‖? = CE(SCD)1−2η
6 ε1−2η
3 C(1/2− η, 3/(1 + η),Ω) , ‖l(2)‖ = (1 + CE)CE(SCD)12 ε‖f‖H1(Ω⊕) .
Proof. Let us start with the upper bound for d(2)(u, v), that is d(2)(u, v) ≤ ‖∇u‖L2(Σ)‖∇v‖L2(Σ) . Let us
assume that u ∈ Hk(Ω) with 1 < k ≤ 2, then ∇u ∈ Hk−1(Ω) and for Sobolev embedding theorem we have∇u ∈ Lp∗(Ω) with p∗ = 6/(5− 2k). Then we apply Holder inequality and (13) as follows,
‖∇u‖L2(Σ) =(∫
Σ
(∇u)2dω) 1
2 ≤(∫
Σ
1 dω) 1
2q(∫
Σ
(∇u)2r dω) 1
2r
≤ |Σ|12q ‖∇u‖L2r(Σ) ≤ (SCD)
12q ε
1q ‖∇u‖L2r(Σ)
where q, r must satisfy 1/q + 1/r = 1. The maximum exponent for which the previous inequality holds trueis p∗ = 2r = 6/(5 − 2k). Then, r = 3/(5 − 2k) and 1/q = 2(k − 1)/3. Denoting with C(k − 1, p∗,Ω) the
21
constant of the Sobolev inequality of the embedding Hk−1(Ω) ⊂ Lp∗(Ω), we have
‖∇u‖L2(Σ) ≤ (SCD)12q ε
1q ‖∇u‖Lp∗ (Σ) ≤ (SCD)
12q ε
1q ‖∇u‖Lp∗ (Ω)
≤ (SCD)12q ε
1qC(k − 1, p∗,Ω)‖∇u‖Hk−1(Ω) ≤ (SCD)
12q ε
1qC(k − 1, p∗,Ω)‖u‖Hk(Ω).
Being u(2) ∈ H 32−η(Ω) and z
(2)ref ∈ H1
∂Ω(Ω⊕), we obtain,
d(u(2), (IΩ⊕ + EΣ)zref) ≤ CE(SCD)1−2η
6 ε1−2η
3 C (1/2− η, 3/(1 + η),Ω) ‖u(2)‖H
32−η(Ω)
‖zref‖H1(Ω⊕) .
Proceeding similarly, using the minimal regularity requirement EΣf ∈ H1(Σ) combined with Holderinequality and Sobolev embeddings, namely H1(Σ) ⊂ L6(Σ), we have
l((IΩ⊕ + EΣ)zref) = −(EΣf, (IΩ⊕ + EΣ)zref)Σ ≤ CE‖EΣf‖L2(Σ)‖zref‖H1(Ω⊕)
≤ CE(SCD)13 ε
23 ‖EΣf‖L6(Σ)‖zref‖H1(Ω⊕) ≤ CE(SCD)
13 ε
23 ‖(IΩ⊕ + EΣ)f‖L6(Ω)‖zref‖H1(Ω⊕)
≤ CE(1 + CE)(SCD)13 ε
23C(1, 6,Ω)‖f‖H1(Ω⊕)‖zref‖H1(Ω⊕) ,
being C(1, 6,Ω) the constant in the Sobolev embedding of H1(Ω) in L6(Ω).
Lemma 4.9. The solution of the reduced problem u(2) satisfies the Property 4.2 with the norm ‖ · ‖? =‖ · ‖
H32−η(Ω)
, for any η > 0 and the inequality (37) is satisfied with the positive constant, uniformly upper
bounded for ε,
‖a(2)‖? = 2CR max[(1 + CE), (1 + CP (Ω))(1 + CE)
√CT (Γ,Ω) , ‖κ‖L∞
√CT (Γ,Ω) , ‖κ‖L∞
√C ′T (Γ,Ω⊕)
].
Proof. We notice that (46) in the strong form reads as
−∆u(2) =(IΩ⊕ + EΣ
)f − κ
(u(2) − U
)δΓ in Ω, u(2) = 0 on ∂Ω.
Then, owing to Theorem 3.11, u(2) ∈ H 32−η(Ω)∩H1
0 (Ω) and using (30) we obtain the following upper bound,
‖u(2)‖H
32−η(Ω)
≤ CR‖(IΩ⊕ + EΣ
)f − κ
(u(2) − U
)δΓ‖
H−12−η(Ω)
. (48)
By the Sobolev embedding H1(Ω) ⊂ H− 12−η(Ω) we have,
‖(IΩ⊕ + EΣ
)f‖
H−12−η(Ω)
≤ (1 + CE)‖f‖H1(Ω⊕) .
From the definition of ‖ · ‖H−
12−η(Ω)
, we have
‖κUδΓ‖H−
12−η(Ω)
= supw∈H
12
+η(Ω)w 6=0
(κUδΓ, w)Ω
‖w‖H
12
+η(Ω)
.
Then the following upper bound holds,
(κUδΓ, w)Ω = (κU,w)Γ ≤ ‖κ‖L∞‖U‖L2(Γ)‖w‖L2(Γ) ≤ ‖κ‖L∞√C ′T (Γ,Ω)‖U‖L2(Λ),|∂D|‖w‖H 1
2+η(Ω)
,
where C ′T (Γ,Ω) is the constant in the trace inequality from L2(Γ) to H12 +η(Ω). Proceeding similarly, we
obtain the following estimate,
‖κu(2)δΓ‖H−
12−η(Ω)
≤ ‖κ‖L∞(Γ)
√C ′T (Γ,Ω)‖u(2)‖L2(Γ) ≤ ‖κ‖L∞(Γ)
√CT (Γ,Ω)C ′T (Γ,Ω)‖u(2)‖H1(Ω).
22
Then, we notice that problem (46) satisfies the following stability property in H1(Ω),
(1 + CP (Ω))−1‖u(2)‖2H1(Ω) ≤ (∇u(2),∇u(2))Ω + (κu(2), u(2))Γ ≤ ((IΩ⊕ + EΣ)f, u(2))Ω + (κU, u(2))Γ
≤ (1 + CE)‖f‖H1(Ω⊕)‖u(2)‖H1(Ω) + ‖κ‖L∞(Γ)
√CT (Γ,Ω)‖U‖L2(Λ),|∂D|‖u(2)‖H1(Ω) ,
that is,
‖u(2)‖H1(Ω) ≤ (1 + CP (Ω)) max(
(1 + CE), ‖κ‖L∞(Γ)
√CT (Γ,Ω)
) (‖f‖H1(Ω⊕) + ‖U‖L2(Λ),|∂D|
).
Combining the upper bounds for each term on the right hand side of (48) we obtain the desired result,
‖u(2)‖? ≤ 2CR max
((1 + CE), ‖κ‖L∞
√C ′T (Γ,Ω⊕), ‖κ‖L∞
√CT (Γ,Ω)C ′T (Γ,Ω⊕)(1 + CP )(1 + CE),
‖κ‖2L∞CT (Γ,Ω)√C ′T (Γ,Ω⊕)
)(‖f‖H1(Ω⊕) + ‖U‖L2(Λ),|∂D|
). (49)
Lemma 4.10. The solution of the reference dual problem, that is to find z(2)ref ∈ H1
∂Ω(Ω⊕) such that
(∇v,∇z(2)ref )Ω⊕ + (κv, z
(2)ref )Γ = j(2)(v), ∀v ∈ H1
∂Ω(Ω⊕), (50)
with j(2)(v) = (e(2), (IΩ⊕ + EΣ)v)Ω, satisfies the following inequality,
(1 + CP (Ω⊕))−1‖z(2)ref ‖
2Ω⊕ ≤ (∇z(2)
ref ,∇z(2)ref )Ω⊕ + (κz
(2)ref , z
(2)ref )Γ = (e(2), (IΩ⊕ + EΣ)z
(2)ref )Ω
≤ (1 + CE)‖e(2)‖L2(Ω)‖z(2)ref ‖H1(Ω⊕) .
As a result, Property 4.2 is satisfied with the norm ‖ · ‖ = ‖ · ‖H1(Ω⊕) and with the constant,
‖a(2)ref ‖ = (1 + CP (Ω⊕))(1 + CE).
4.3. Analysis of the modeling error of the transmission conditions (assumption A3)
The interface conditions between Σ and Ω⊕ must be adapted to the topological model reduction of theproblem on Σ. This is achieved by averaging the solution u⊕ on cross sections of the interface ∂D(s) beforeenforcing the interface condition between u⊕ and U . The error that arises in this process corresponds toneglect the fluctuations of u⊕ at the interface, as stated in assumption A3. In this case, the reference problem
is (46), with solution u(3)ref = u(2), while the reduced problem is (8) so that u(3) = u, being u the solution
of the final reduced model on Ω. For the sake of clarity, as was done in the previous cases, we report thereference and the reduced problems here:
find u(2) ∈ H10 (Ω) : (∇u(2),∇v)Ω + (κu(2), v)Γ = ((IΩ⊕ + EΣ)f, v)Ω + (κU, v)Γ , ∀v ∈ H1
0 (Ω) , (51)
find u ∈ H10 (Ω) : (∇u,∇v)Ω + (κu, v)Λ,|∂D| = ((IΩ⊕ + EΣ)f, v)Ω + (κU, v)Λ,|∂D| , ∀v ∈ H1
0 (Ω) . (52)
The modeling error is easily defined as e(3) = u(3)ref − u(3) = u(2) − u ∈ H1
0 (Ω) for any u, v ∈ H10 (Ω).
Provided that U is uniformly extended from H1(Λ) to H1(Σ) the difference operator between (51) and (52)is
d(3)(u, v) = (κu, v)Γ − (κu, v)Λ,|∂D| = (κ(I − (·))u, v)Γ , (53)
l(3)(v) = (κU, v)Γ − (κU, v)Λ,|∂D| = 0 . (54)
23
Lemma 4.11. Under the assumption that u ∈ H10 (Ω) ∩ H 3
2 (Σ), the operators d(3)(·, ·) and l(3)(·) satisfyProperty 4.1 with the norms ‖ · ‖? = ‖ · ‖ = ‖ · ‖H1(Ω) and with constants
‖d(3)‖? = ‖κ‖L∞√CT (Γ,Ω)
√CSε , ‖l(3)‖ = 0 .
In particular, if we take into account the dependence of CT (Γ,Ω) on ε, using Lemma 3.4, we obtain that
‖d(3)‖? = O(ε1−
12(2−δ)
).
Proof. Since u ∈ H 32 (Σ), we have that u|D(s) ∈ H1(D(s)) for any s ∈ (0, S). Then we can apply Stekloff
inequality on any cross section of Σ. Combining it it with Cauchy-Schwarz and trace inequalities, we obtain,
d(3)(u, zref) = (κ(I − (·))u, zref)Γ ≤ ‖κ‖L∞√CSε
(∫Λ
∫D(s)
(∇Du)2 dσ ds
) 12
‖zref‖L2(Γ)
≤ ‖κ‖L∞√CSε‖∇u‖L2(Σ)‖zref‖L2(Γ) ≤ ‖κ‖L∞
√CT (Γ,Ω)
√CSε‖u‖H1(Ω)‖zref‖H1(Ω).
Lemma 4.12. The solution u(3) of problem (52) satisfies the following inequality
‖u(3)‖H1(Ω) ≤ (1 + CP (Ω)) max(
(1 + CE), ‖κ‖L∞√CT (Γ,Ω)
) (‖f‖H1(Ω⊕) + ‖U‖L2(Λ),|∂D|
).
As result, Property 4.2 is satisfied with the norm ‖ · ‖? = ‖ · ‖H1(Ω) and with the constant
‖a(3)‖? = (1 + CP (Ω)) max(
(1 + CE), ‖κ‖L∞√CT (Γ,Ω)
).
Proof. From problem (52) we obtain,
(1 + CP (Ω))−1 ‖u(3)‖2H1(Ω) ≤ (∇u,∇u)Ω + (κu, u)Λ,|∂D|
≤ ‖(IΩ⊕ + EΣ)f‖L2(Ω)‖u(3)‖L2(Ω) + ‖κ‖L∞‖U‖L2(Λ),|∂D|‖u‖L2(Λ),|∂D|
≤ (1 + CE)‖f‖L2(Ω⊕)‖u(3)‖H1(Ω) + ‖κ‖L∞√CT (Γ,Ω)‖U‖L2(Λ),|∂D|‖u(3)‖H1(Ω),
from which it follows that
‖u(3)‖H1(Ω) ≤ (1 + CP (Ω)) max(
1 + CE , ‖κ‖L∞√CT (Γ,Ω)
) (‖f‖H1(Ω⊕) + ‖U‖L2(Λ),|∂D|
).
Consequently,
‖a(3)‖? = (1 + CP (Ω)) max(
1 + CE , ‖κ‖L∞√CT (Γ,Ω)
).
Lemma 4.13. The solution of the reference dual problem, that is to find z(3)ref ∈ H1
0 (Ω) such that
(∇v,∇z(3)ref )Ω + (κv, z
(3)ref )Γ = j(3)(v), ∀v ∈ H1
0 (Ω), (55)
with j(3)(v) = (e(3), v)Ω, satisfies the following inequality
‖z(3)ref ‖H1(Ω) ≤ (1 + CP (Ω))‖e(3)‖L2(Ω).
As result, Property 4.2 is satisfied with the norm ‖ · ‖ = ‖ · ‖H1(Ω) and the constant
‖a(3)ref ‖ = 1 + CP (Ω).
24
4.4. Conclusions of the model error analysis
In the previous sections we have analyzed three different components of the model error. In particular,we recall that
e(1) = u(1)ref − u
(1) = u − U ∈ H1(Σ)
e(2) = u(2)ref − u
(2) = (IΩ⊕ + EΣ)u⊕ − u(2) ∈ H10 (Ω)
e(3) = u(3)ref − u
(3) = u(2) − u ∈ H10 (Ω)
As a result, the total model error, on Σ and Ω respectively can be straightforwardly decomposed as
eΣ + eΩ = (u − U) + ((IΩ⊕ + EΣ)u⊕ − u) = e(1) + e(2) + e(3) ,
with eΣ = e(1) ∈ H1(Σ) and eΩ = e(2) + e(3) ∈ H10 (Ω). Then, for a suitable choice of the model error output
functionals used in the reference dual problems, if Properties (4.1) and (4.2) are satisfied, the followinginequality holds true,
j(k)(e) = ‖e(k)‖L2 ≤(‖l(k)‖ + ‖d(k)‖?‖a(k)‖?‖F‖X′
)‖a(k)
ref ‖, k = 1, 2, 3.
For eΣ, if u⊕ ∈ H1∂Ω(Ω⊕) ∩H2(Ω⊕), we have
‖e(1)‖L2(Σ) ≤(‖l(1)‖ + ‖d(1)‖?‖a(1)‖?‖F‖X′
)‖a(1)
ref ‖ = ‖l(1)‖‖a(1)ref ‖ = O(ε
12 ) , (56)
because ‖l(1)‖ = O(ε12 ) for Lemma 4.6, ‖d(1)‖? = 0 for Lemma 4.5 and ‖a(1)
ref ‖ is uniformly bounded with
respect to ε. For eΩ, if u ∈ H10 (Ω) ∩H 3
2 (Σ), from Lemmas 4.8, 4.9, 4.10 we obtain
‖e(2)‖L2(Ω) ≤(‖l(2)‖ + ‖d(2)‖?‖a(2)‖?
(‖f‖H1(Ω⊕) + ‖U‖L2(Λ),|∂D|
))‖a(2)
ref ‖ = O(ε1−2η
3 ) , (57)
and from Lemmas 4.11, 4.12, 4.13 we have
‖e(3)‖L2(Ω) ≤ ‖d(3)‖?‖a(3)‖?(‖f‖H1(Ω⊕) + ‖U‖L2(Λ),|∂D|
)‖a(3)
ref ‖ = O(ε12 ) . (58)
By combining the previous upper bounds for ‖e(2)‖L2(Ω) and ‖e(3)‖L2(Ω) we obtain the following estimatefor ‖eΩ‖L2(Ω)
‖eΩ‖L2 ≤ ‖l(2)‖‖a(2)ref ‖ +
(‖d(2)‖?‖a(2)‖?‖a(2)
ref ‖ + ‖d(3)‖?‖a(3)‖?‖a(3)ref ‖
) (‖f‖H1(Ω⊕) + ‖U‖L2(Λ),|∂D|
)which entails that ‖eΩ‖L2(Ω) = O(ε
1−2η3 ) as ε→ 0.
Remark 4.14. The model error analysis relies on two additional regularity assumptions, u⊕ ∈ H2(Ω⊕) and
u ∈ H 32 (Σ). These are technical assumptions required to use the Stekloff inequality (39). The former one is
more restrictive because EΣu⊕ ∈ H32 (Ω⊕) requires u⊕ ∈ H2(Ω⊕) since Lemma 4.3 works for Sobolev spaces
with integer index. We remark that none of the constants appearing in (56), (57) and (58) involve neither‖u⊕‖H2(Ω⊕) nor ‖u‖H3/2(Σ).
Furthermore, we observe that these regularity assumptions are local regularity properties on subregionsof Ω that do not cross the surface Γ. As a consequence, these properties can be justified with classicalregularity results for elliptic equations with regular coefficients and on domains with smooth boundary, seefor example Theorem 3.10 of [15] and references therein. We report below a sketch of the proofs, althoughwe do not claim to be fully rigorous.
For the former assumption, we consider problem (1) and we fix u ∈ H1(Σ). Then u⊕ solves a Poissonproblem with Robin boundary conditions on the inner boundary of Ω⊕ and homogeneous Dirichlet conditions
25
on the outer boundary, which are disjoint. Given a C2 regular domain, a right hand side f ∈ L2(Ω⊕) and a
forcing term of the Robin condition that is g = κu ∈ H12 (Γ), we conclude that u⊕ ∈ H2(Ω⊕).
For the latter assumption, we observe from equation (8) that the restriction of u to Σ solves the followingproblem in the weak sense,
−∆u = f in Σ,
−∇u · n = κEΓ(u− U) on Γ,
−∇u · n = 0 on ∂Σ \ Γ.
Given the standard H1-regularity of Theorem 3.3 we have EΓ(u − U) ∈ H 12 (Γ). If this function smoothly
vanishes approaching ∂Σ\Γ, then this problem is a Neumann problem with H12 -regular data on the boundary.
As a result, standard H2-regularity also apply to the restriction of u on Σ .
5. Numerical experiments
The purpose of this section is twofold. First we introduce the basic aspects of the numerical approximationof the problem by means of finite elements. In this context, we validate the numerical solver on a simpleproblem for which the solution is available explicitly. Second, we use the numerical method to calculatea local error estimator of the modeling error. This is a preliminary attempt of a posteriori model erroranalysis, that is used here for a qualitative investigation of the spatial distribution of the model error.
5.1. Finite element approximation
Let us consider a quasi-uniform partition T hΩ of Ω and an admissible partition T hΛ of Λ with comparablecharacteristic size, denoted by h, and let Vh = V Ω
h × V Λh ⊂ V be continuous k1, k2-order Lagrangian finite
element spaces defined on T hΩ , T hΛ respectively. The numerical approximation of the variational formulation(12) consists of finding Uh ∈ Vh solution of
A(Uh,Vh) = F(Vh) ∀Vh ∈ Vh. (59)
We notice that in problem (59) it is implicitly assumed that numerical integration is performed exactly.
In practice, the average operator (·) is approximated by means of numerical quadrature. The effect of thelatter approximation shall be analyzed in a future development of this work. Owing to this assumption, thediscretization method is consistent and conformal with (12).
For the numerical discretization, we first address the discrete counterparts of (BNB1)-(BNB2), namely,
∃αh > 0 : infWh∈Vh
supVh∈Vh
A(Wh,Vh)
|||Wh||| |||Vh|||≥ α , (BNB1h)
∀Vh ∈ Vh :(A(Wh,Vh) = 0 ∀Wh ∈ Vh
)⇒ Vh = 0 , (BNB2h)
which are not necessarily implied by the corresponding continuous properties. As shown in Proposition 2.21of [15], if the test and the search space of (59) are the same, namely Wh ≡ Vh then (BNB1h) is equivalent to(BNB2h). As a consequence of that, we can verify either (BNB1h) or (BNB2h). We prove the latter using(21) that holds true because of the consistency and the conformity of the method. Owing to this property,A(Vh,Vh) = 0 implies that
‖vh‖H1(Ω) = 0, |Vh|H1(Λ),|D| = 0, ‖vh − Vh‖L2(Λ),|∂D| = 0,
which is equivalent to vh = 0 and |Vh|H1(Λ),|D| = 0 with ‖Vh‖2L2(Λ),|∂D| = 0, that is Vh = 0.
We exploit the consistency and conformity of the discretization method combined with (BNB1h) andLemma 3.7, in order to prove that Uh satisfies a Cea-type inequality ( [15] [Lemma 2.28]),
|||U − Uh||| ≤(
1 +|||A|||α
)inf
vh∈V Ωh ,Vh∈V
Λh
(‖u− vh‖H1(Ω) + ‖U − Vh‖H1(Λ),|D|
). (60)
26
The convergence of the finite element method follows from (60) combined with approximation propertiesof the finite element spaces. For the latter property, we exploit the additional regularity of the solutionin Ω proved in Theorem 3.11 and the fact that the solution U on Λ is in H2(Λ). The regularity of Udescends from the standard properties of elliptic operators in one space dimension, as the right hand sideof equation (11b) belongs to L2(Λ). From now on, let a . b be equivalent to the inequality a ≤ Cb whereC is a generic constant, possibly dependent on Ω, Λ but independent of the parameters of the problem.Concerning the solution u in Ω, let πh be the Scott-Zhang interpolation operator from W l,q(Ω) ∩H1
0 (Ω) toV Ωh with 1 ≤ q ≤ ∞ and 0 ≤ l ≤ k1 + 1, with the additional constraint l ≥ 1/q when q > 1. Then, the
following interpolation estimate holds true in the norm of W t,q(Ω) with t ≤ l (see for example [15, Lemma1.130])
‖v − πhv‖W t,q(Ω) . hl−t|v|W l,q(Ω).
The estimate above applies to the problem at hand, knowing that u ∈ H32−η(Ω) ∩ H1
0 (Ω), with t = 1,l = 3
2 − ε, q = 2, k1 = 1, obtaining
infvh∈V Ω
h
‖u− vh‖H1(Ω) . h12−η‖u‖
H32−η(Ω)
.
For the solution U on Λ and k2 = 1, the standard finite element approximation estimate ensures that
infVh∈V Λ
h
‖U − Vh‖H1(Λ) . h‖U‖H2(Λ).
Therefore, combining (60) and the previous inequalities for piecewise affine approximation, we obtain
|||U − Uh||| . h12−η‖u‖
H32−η(Ω)
+ h‖U‖H2(Λ).
We have tested the previous convergence results by means of numerical experiments based on a problemfor which the analytical solution is known. Precisely, we consider Ω = (−1, 1)3 ⊂ R3 and Σ is the cylinderwith constant circular cross section of radius R = 0.25 and centerline Λ = (x, 0, 0), x ∈ (−1, 1). We assumeU = 1, therefore the problem reduces to find only the solution u in Ω. Concerning the other parameters, wechoose f = 0 and κ = 0.1. With appropriate boundary conditions, the exact solution ue of the problem canbe obtained by uniform extension along the x-coordinate the 2D solution given in [26], which is
u2De (y, z) =
U κ
1+κ
(1−R ln r
R (y, z),)
r(y, z) > R,
U κ1+κ , r(y, z) ≤ R,
where r(y, z) is the Euclidean distance from the origin. In particular in our case ue(x, y, z) = u2De (y, z).
Starting from a quasi-uniform triangulation of Ω and applying different uniform subdivisions along the yand z axes (denoted as # sub.), we calculate the discretization error eh = ue−uh and the rate of convergencewith respect to the H1- norm. The numerical results reported in Figure 2 agree with the theoretical estimate.
5.2. A posteriori model error analysis
Starting from the definitions of the difference operators d(k)(·, ·) and l(k)(·) k = 1, 2, 3, defined in Sections4.1, 4.2 and 4.3, we compute a given model error output functional, through the error representation formula(35). In particular, we localize the error on the elements of the triangulation T hΩ of Ω. To perform this task,we interpret (35) as the sum of residuals weighed with the dual solution.
First, we introduce the local error estimator using the abstract setting defined at the beginning of Section4. Let Xh ⊂ X be a suitable finite element space of dimension N . We denote with uh and zh,ref the discretesolutions of the reduced problem and reference dual problem, respectively. Considering the Lagrangian nodal
basis φi ⊂ Xh, we define the vector of residuals, ρ = ρiNi=1, with ρi = l (φi) − d (uh) (φi). Accordingto (35) the local weights, ωi, correspond to the degrees of freedom of the discrete dual solution, namely
zh =∑Ni=1 ωiφi. Let 〈·, ·〉 be the Euclidean scalar product in RN . The model error output functional is then
j(e) = 〈ρ,ω〉 and the components of the local error estimator η ∈ RN are ηi = ρiωi. The vector η can be27
# sub. ‖eh‖H1(Ω) Rate
8 1.77839e-02 -16 1.21790e-02 0.5461832 8.63759e-03 0.4956964 6.10750e-03 0.50005
Figure 2. On the top we show a visualization of the exact solution ue and the numericalsolution uh, superposed to the quasi-uniform mesh used for the convergence test. On thebottom we report the approximation error and the corresponding convergence rate.
represented on Ω as a finite element function ηh =∑Ni=1 ηiφi. Each nodal component of η represents the
localization of the error on T hΩ .Using this general approach we explicitly calculate the estimators η1, η2, η3 of the model error of Sections
4.1, 4.2 and 4.3. To this aim, we must compute the residuals ρ(k) and the weights ω(k) for k = 1, 2, 3. For
the weights, we recall that z(1)ref ∈ H1(Σ), z
(2)ref ∈ H1
∂Ω(Ω⊕), z(3)ref ∈ H1
0 (Ω). Then, we introduce V Σh , V
Ω⊕h ,
V Ωh the finite element spaces over Σ, Ω⊕, Ω, respectively. Furthermore, we assume that the following direct
sum decomposition holds V Ωh = V Σ
h
⊕V
Ω⊕h . In practice, we consider the particular case of a mesh T hΩ that
nodally conforms with the interface Γ and we assign all the nodes on the interface to VΩ⊕h . According to
this notation the discrete reference dual solutions are z(1)h,ref ∈ V Σ
h , z(2)h,ref ∈ V
Ω⊕h , z
(3)h,ref ∈ V Ω
h . However, for
computational convenience, we define the all the weight functions on the finite element space V Ωh ⊂ H1
0 (Ω).
Let NΩh = dim
(V Ωh
)be the degrees of freedom of such space and let φi
NΩh
i=1 be the corresponding finite
element basis. As a result, all the vectors of weights and residuals belong to RNΩh . Such weights are obtained
from the dual solutions by means of the following extensions,
ω(1)i =
zi : z
(1)h,ref =
∑NΣh
i=1 ziφi i = 1, . . . , NΣh
0 i = NΣh + 1, . . . , NΩ
h
(61a)
ω(2)i =
0 i = 1, . . . , NΣ
h
zi : z(2)h,ref =
∑NΩ⊕h
i=1 ziφi i = NΣh + 1, . . . , NΩ
h
(61b)
ω(3)i = zi : z
(3)h,ref =
∑NΩh
i=1 ziφi i = 1, . . . , NΩh . (61c)
28
We then build the residuals ρ(k) for k = 1, 2, 3, depending on the discrete reduced solutions uh and Uh,computed by means of (59). Using the definitions of (43), (47), (53), for i = 1, . . . , NΩ
h the nodal componentsof the residuals are,
ρ(1)i = l(1)(φi)− d(1)(Uh, φi) =((I − (·))g, φi)Σ + (κ(I − (·))uh, φi)Γ, (62a)
ρ(2)i = l(2)(φi)− d(2)(uh, φi) =(∇uh,∇φi)Σ − (EΣf, φi)Σ, (62b)
ρ(3)i = l(3)(φi)− d(3)(uh, φi) =− (κ(I − (·))uh, φi)Γ, (62c)
Finally, we combine the weights and the residuals in order to compute the local estimators,
η(1)i = ω
(1)i ρ
(1)i =ω
(1)i
[((I − (·))g, φi)Σ + (κ(I − (·))uh, φi)Γ
], (63a)
η(2)i = ω
(2)i ρ
(2)i =ω
(2)i [(∇uh,∇φi)Σ − (EΣf, φi)Σ] , (63b)
η(3)i = ω
(3)i ρ
(3)i =ω
(3)i
[−(κ(I − (·))uh, φi)Γ
]. (63c)
Remark 5.1. We observe that when the mesh nodally conforms with Γ, the advantage in terms of compu-tational cost of the reduced versus the full (reference) problem almost vanishes. However, we remark thatthis is a particular case specifically designed to simplify the computations presented below and practicallyfeasible only when the radius of Σ, R, is comparable with the diameter of Ω.
For the computation of the local model error estimator, we consider the 3D domain Ω = (−1, 1)2 ×(−0.51, 0.51) and the 1D segment Λ from (−0.51, 0, 0) to (0.51, 0, 0). We define on Ω a a quasi-uniformregular mesh, with characteristic length h = 1/32, for a total of 354’753 tetrahedra. The 1D domain isdiscretized with 1281 points. The parameters of the problem are chosen for simplicity as R = 0.25, k = 1,f = 1 and g = 1.
The discrete solutions uh and Uh are computed using piecewise linear finite elements. We calculate theresiduals ρ(k) by means of (62). We then address the discretization of the reference dual problems (44), (50),(55). To obtain computable solutions, the model error output functionals must not depend on the erroritself. We use instead the following definitions,
j(1)(v) =
∫Σ
vdΣ , j(2)(v) =
∫Ω
(IΩ⊕ + EΣ
)vdΩ , j(3)(v) =
∫Ω
vdΩ . (64)
These output functionals provide the mean value of the error components over Σ and Ω, respectively. Given
the mesh T hΩ that is nodally conforming with the surface Γ, we define the spaces V Σh , V
Ω⊕h , V Ω
h using
piecewise linear elements, we solve the problems (44), (50), (55) and we compute the weights w(k) using(61). According to (63) the nodal values of the local estimators are the combination of residual and weights.We represent them as piecewise linear finite element functions on Ω.
In what follows, all the results refer to 3D functions in Ω, but the meaningful data are strictly inside thedomain, because of the homogeneous Dirichlet condition on his boundary. Therefore, in order to successfullyrepresent the information of the plots, we show only a 2D slice embedded in the 3D domain. The boundingbox of Ω is also shown in the plots.
For the discussion of Figure 3 we start from the residual ρ(1), in particular ((I − (·))g, φi)Σ. This termmeasures the error due to the average on the cross section D of the forcing term. Since in the simulation ofFigure 3, we have used g = 1 this residual should vanish. A different test using g = 1 + 3y is addressed inFigure 4 (left column). The comparison among the two cases conforms that, g is variable on D, the residualreproduces the variation of g.
Similar considerations apply to the residual ρ(2). It represents the model error due to extending the 3Dproblem into Σ. However, even though the magnitude of this residual is the largest among all ρ(k), k = 1, 2, 3,we notice that it is combined with ω(2), which according to (61b) vanishes in the interior of Σ. As a result,the estimator η(2) turns out to be rather small and localized in the neighborhood of the interface.
We also discuss terms on the interface (κ(I − (·))u, φi)Γ that appear in ρ(1) and ρ(3). These residualsdepend on the function f . Precisely, if f is constant, then also the solution u does not vary along ∂D and
29
ρ(1)h ρ
(2)h ρ
(3)h
ω(1)h ω
(2)h ω
(3)h
η(1)h η
(2)h η
(3)h
Figure 3. Analysis of the residuals ρ(k)h , the weights ω
(k)h and the local estimators η
(k)h ,
represented as piecewise linear functions on the mesh T hΩ .
u(s) = u(s). Therefore, the residuals ρ(1) and ρ(3) become relevant especially when f is not constant aroundthe inclusion. The comparison of these residuals in the case f = 1 and f = 1 + y, shown in Figure 4 (rightcolumn), confirms the expected behavior.
30
Case g = 1 Case f = 1
((I − (·))g, φi)Σ ((I − (·))u, φi)Γ
Case g = 1 + 3y Case f = 1 + y
((I − (·))g, φi)Σ ((I − (·))u, φi)Γ
Figure 4. On the left column we show the comparison of (g, φi)Σ and ((·))g, φi)Σ in thecase g = 1 and g = 1 + 3y. On the right, we analyze the comparison of (u, φi)Γ and (u, φi)Γ
in the case f = 1 and f = 1 + y.
Finally, we consider a test case with a thinner inclusion, namely a cylinder with radius R = 0.1. Thedomains Ω and Λ as well as the parameters k, f and g are the same as in the previous case shown in Figure3. The computational mesh has been slightly refined in the neighborhood of the cylinder, to comply withits size. The visual analysis of the error estimator (not shown here) confirms that the spatial distribution ofthe error is the same as the one of Figure 3. In Table 1 we show a quantitative comparison of the two cases.As expected, the results confirm that the model error decreases with the size of the inclusion.
6. Conclusions
This work illustrates the foundations of a multiscale method for solving partial differential equationson bulk domains with embedded network-shaped cylindrical inclusions. The method consists of applyinga model reduction technique first, aiming at transforming the original problem into a simpler one. Suchproblem features equations in a bulk domain coupled with a collection of one-dimensional problems. Aftershowing the well posedness of the reduced problem, we have approximated it by means of finite elements andproved the convergence of the discretization scheme. Finally, we have pursued the analysis of the modelingerror, namely the difference between the solutions of the original problem and the simplified one. We have
31
j(1)(e) =∑i η
(1)i j(2)(e) =
∑i η
(2)i j(3)(e) =
∑i η
(3)i
R = 0.25 2e-04 -1e-02 -1e-05R = 0.1 9e-07 -3e-05 -1e-06
η(1) η(2) η(3)
mini maxi mini maxi mini maxi
R = 0.25 -8e-21 4e-08 -1e-05 9e-07 -1e-25 3e-08R = 0.1 -1e-11 4e-10 -8e-09 1e-07 -3e-11 8e-10
Table 1. Variation of the error output functionals j(k)(e) and of the error estimators η(k)
when the radius of the inclusion Σ decreases from R = 0.25 to R = 0.1.
shown that for infinitesimally narrow inclusions, representing for example a bulk material perfused by narrowchannels, the reduced model converges to the original one. Although realistic applications require to addressmore complicated equations, which have not been fully analyzed yet in this context, we believe that thesound mathematical properties proved here strengthen the significance of this approach to applications.However, there are still many open questions to be addressed. For example, this work should be extendedto other types of interface conditions among the bulk and the inclusions, addressing for example Dirichlettype constraints. In the same spirit, we are studying the method for partial differential equations in mixedform, in order to better model flow problems. Important and pressing questions also arise at the levelof numerical discretization and solvers. For example, the role of quadrature in the approximation of theaveraging operators on the global accuracy of the approach is still unexplored. At the level of numericalsolver, efficient solution methods and preconditioners for coupled partial differential equations on embeddeddomains must be investigated.
References
[1] Serge Alinhac and Patrick Gerard. Pseudo-differential operators and the Nash-Moser theorem, volume 82 of Graduate
Studies in Mathematics. American Mathematical Society, Providence, RI, 2007. Translated from the 1991 French originalby Stephen S. Wilson.
[2] Mario Arioli and Michele Benzi. A finite element method for quantum graphs. IMA J. Numer. Anal., 38(3):1119–1163,
2018.[3] Gregory Berkolaiko, Robert Carlson, Stephen A. Fulling, and Peter Kuchment. Quantum graphs and their applications,
volume 415 of Contemporary Mathematics. American Mathematical Society, Providence, RI, 2006.
[4] S. Bertoluzza, A. Decoene, L. Lacouture, and S. Martin. Local error estimates of the finite element method for an ellipticproblem with a Dirac source term. Numerical Methods for Partial Differential Equations, 34(1):97–120, 2018.
[5] T.R. Blake and J.F. Gross. Analysis of coupled intra- and extraluminal flows for single and multiple capillaries. MathematicalBiosciences, 59(2):173–206, 1982.
[6] Wietse Boon, Jan Nordbotten, and Jon Vatne. Functional analysis and exterior calculus on mixed-dimensional geometries.
Technical report, arXiv, Cornell University Library, 2018. arXiv:1710.00556v3.[7] Malte Braack and Alexandre Ern. A posteriori control of modeling errors and discretization errors. Multiscale Model.
Simul., 1(2):221–238, 2003.[8] Susanne C. Brenner. Poincare-Friedrichs inequalities for piecewise H1 functions. SIAM Journal on Numerical Analysis,
41(1):306–324, 2003.
[9] L. Cattaneo and P. Zunino. A computational model of drug delivery through microcirculation to compare different tumor
treatments. International Journal for Numerical Methods in Biomedical Engineering, 30(11):1347–1371, 2014.[10] L. Cattaneo and P. Zunino. Computational models for fluid exchange between microcirculation and tissue interstitium.
Networks and Heterogeneous Media, 9(1):135–159, 2014.[11] D. Cerroni, F. Laurino, and P. Zunino. Mathematical analysis, finite element approximation and numerical solvers for the
interaction of 3d reservoirs with 1d wells. GEM - International Journal on Geomathematics, 10(1), 2019.
[12] C. D’Angelo. Multi scale modelling of metabolism and transport phenomena in living tissues, PhD Thesis. EPFL, Lausanne,2007.
[13] C. D’Angelo. Finite element approximation of elliptic problems with Dirac measure terms in weighted spaces: applications
to one- and three-dimensional coupled problems. SIAM Journal on Numerical Analysis, 50(1):194–215, 2012.[14] C. D’Angelo and A. Quarteroni. On the coupling of 1d and 3d diffusion-reaction equations. Application to tissue perfusion
problems. Mathematical Models and Methods in Applied Sciences, 18(8):1481–1504, 2008.
32
[15] Alexandre Ern and Jean-Luc Guermond. Theory and practice of finite elements, volume 159 of Applied MathematicalSciences. Springer-Verlag, New York, 2004.
[16] Julian Fernandez Bonder and Julio D. Rossi. Asymptotic behavior of the best Sobolev trace constant in expanding and
contracting domains. Commun. Pure Appl. Anal., 1(3):359–378, 2002.[17] G.J. Fleischman, T.W. Secomb, and J.F. Gross. The interaction of extravascular pressure fields and fluid exchange in
capillary networks. Mathematical Biosciences, 82(2):141–151, 1986.
[18] G.J. Flieschman, T.W. Secomb, and J.F. Gross. Effect of extravascular pressure gradients on capillary fluid exchange.Mathematical Biosciences, 81(2):145–164, 1986.
[19] I. Gansca, W. F. Bronsvoort, G. Coman, and L. Tambulea. Self-intersection avoidance and integral properties of generalizedcylinders. Comput. Aided Geom. Design, 19(9):695–707, 2002.
[20] David Gilbarg and Neil S. Trudinger. Elliptic partial differential equations of second order. Classics in Mathematics.
Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.[21] Ingeborg Gjerde, Kundan Kumar, Jan M. Nordbotten, and Barbara Wohlmuth. Splitting method for elliptic equations
with line sources. ESAIM: Mathematical Modelling and Numerical Analysis, pages 1–26, 2019. Accepted article in print.
[22] W. Gong, G. Wang, and N. Yan. Approximations of elliptic optimal control problems with controls acting on a lowerdimensional manifold. SIAM Journal on Control and Optimization, 52(3):2008–2035, 2014.
[23] T. Koch, K. Heck, N. Schrder, H. Class, and R. Helmig. A new simulation framework for soilroot interaction, evaporation,
root growth, and solute transport. Vadose Zone Journal, 17(1), 2018.[24] T. Koppl, E. Vidotto, and B. Wohlmuth. A local error estimate for the Poisson equation with a line source term. In
Numerical Mathematics and Advanced Applications ENUMATH 2015, pages 421–429. Springer, 2016.
[25] T. Koppl and B. Wohlmuth. Optimal a priori error estimates for an elliptic problem with Dirac right-hand side. SIAMJournal on Numerical Analysis, 52(4):1753–1769, 2014.
[26] Tobias Koppl, Ettore Vidotto, Barbara Wohlmuth, and Paolo Zunino. Mathematical modeling, analysis and numericalapproximation of second-order elliptic problems with inclusions. Mathematical Models and Methods in Applied Sciences,
28(05):953–978, 2018.
[27] M. Kuchta, M. Nordaas, J.C.G. Verschaeve, M. Mortensen, and K.-A. Mardal. Preconditioners for saddle point systemswith trace constraints coupling 2d and 1d domains. SIAM Journal on Scientific Computing, 38(6):B962–B987, 2016.
[28] Miroslav Kuchta, Kent-Andre Mardal, and Mikael Mortensen. Preconditioning trace coupled 3d-1d systems using fractional
Laplacian. Numer. Methods Partial Differential Equations, 35(1):375–393, 2019.[29] J. R. Kuttler and V. G. Sigillito. An inequality of a Stekloff eigenvalue by the method of defect. Proc. Amer. Math. Soc.,
20:357–360, 1969.
[30] M. Lesinigo, C. D’Angelo, and A. Quarteroni. A multiscale Darcy-Brinkman model for fluid flow in fractured porous media.Numerische Mathematik, 117(4):717–752, 2011.
[31] M. Nabil, P. Decuzzi, and P. Zunino. Modelling mass and heat transfer in nano-based cancer hyperthermia. Royal Society
Open Science, 2(10), 2015.[32] M. Nabil and P. Zunino. A computational study of cancer hyperthermia based on vascular magnetic nanoconstructs. Royal
Society Open Science, 3(9), 2016.[33] D. Notaro, L. Cattaneo, L. Formaggia, A. Scotti, and P. Zunino. A Mixed Finite Element Method for Modeling the Fluid
Exchange Between Microcirculation and Tissue Interstitium, pages 3–25. Springer International Publishing, 2016.
[34] L. E. Payne and H. F. Weinberger. An optimal Poincare inequality for convex domains. Arch. Rational Mech. Anal.,5:286–292 (1960), 1960.
[35] Donald W. Peaceman. Interpretation of well-block pressures in numerical reservoir simulation with nonsquare grid blocks
and anisotropic permeability. Society of Petroleum Engineers journal, 23(3):531–543, 1983.[36] D.W. Peaceman. Interpretation of well-block pressures in numerical reservoir simulation. Soc Pet Eng AIME J, 18(3):183–
194, 1978.
[37] L. Possenti, G. Casagrande, S. Di Gregorio, P. Zunino, and M.L. Costantino. Numerical simulations of the microvascularfluid balance with a non-linear model of the lymphatic system. Microvascular Research, 122:101–110, 2019.
[38] L. Possenti, S. di Gregorio, F.M. Gerosa, G. Raimondi, G. Casagrande, M.L. Costantino, and P. Zunino. A computa-
tional model for microcirculation including Fahraeus-Lindqvist effect, plasma skimming and fluid exchange with the tissueinterstitium. International Journal for Numerical Methods in Biomedical Engineering, 35(3), 2019.
[39] A. Quarteroni, A. Veneziani, and C. Vergara. Geometric multiscale modeling of the cardiovascular system, between theoryand practice. Computer Methods in Applied Mechanics and Engineering, 302:193–252, 2016.
[40] G. Raimondi. Computational models for root water uptake. Master’s thesis, Politecnico di Milano, 2017.
[41] S. A. Sauter and R. Warnke. Extension operators and approximation on domains containing small geometric details.East-West J. Numer. Math., 7(1):61–77, 1999.
[42] T. Secomb, R. Hsu, E. Park, and M. Dewhirst. Green’s function methods for analysis of oxygen delivery to tissue by
microvascular networks. Annals of Biomedical Engineering, 32(11):1519–1529, 2004.[43] Michael Solomyak. On approximation of functions from Sobolev spaces on metric graphs. J. Approx. Theory, 121(2):199–
219, 2003.
[44] A.-K. Tornberg and B. Engquist. Numerical approximations of singular source terms in differential equations. Journal ofComputational Physics, 200(2):462–488, 2004.
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