Two-gird methods for semilinear elliptic interfaceproblems by immersed finite element methods
Yanping Chen
South China Normal University
Workshop on Modeling and Simulation of Interface-related Problems, 30 Apr - 3 May, 2018, Singapore
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Overview
1 Introduction
2 Immersed interface finite element space
3 Immersed interface finite element method for semi-linear interfaceproblems
4 Two-grid algorithms for semi-linear interface problems
5 Conclusions and future work
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1 Introduction
2 Immersed interface finite element space
3 Immersed interface finite element method for semi-linear interfaceproblems
4 Two-grid algorithms for semi-linear interface problems
5 Conclusions and future work
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Introduction
Interface problems often occur when the computational domainconsists of two different media, which have applications in many physicaland engineering problems, such as fluid dynamics, materials science,electromagnetics, seismo-acoustics.
In many applications, a simulation domain is often formed by severalmaterials separated by curves or surfaces from each other, and this oftenleads to differential equations on irregular domain consisting of the usualboundary condition, plus jump conditions across the material interfacerequired by pertinent physics. Solving interface problems efficiently iscritical, as:
discontinuity across the interfaces;
interfaces with complicated geometries or moving with time;
low global regularity.
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Early works
Finite element methods using body fitting grids:
Ivo Babuska [Computing, 1970] introduced an equivalentminimization problem to handle the jump interface condition usingfitted finite element method.
James H. Bramble and J. Thomas King [Adv. Comput. Math., 1996]derived a finite element method in which the smooth boundary andinterface of the problem are approximated by polygonal domain andinterface.
Zhiming Chen and Jun Zou [Numer. Math., 1998] shown that for C 2
interfaces in 2D convex polygonal domains, the linear finite elementapproximation uh has suboptimal standard error estimates of ordersO(h|logh|1/2) and O(h2|logh|1/2) in H1 and L2 norms, respectively.
Sinha and Deka [Appl. Numer. Math., 2009] studied linear finiteelement approximation of semi-linear elliptic interface problems intwo-dimensional convex polygonal domains.
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Hui Xie, Zhilin Li, and Zhonghua Qiao, [Int. J. Numer. Anal. Model.,2013] studied a finite element method for elasticity interface problemsbased on a Cartesian mesh with local modifications.... ...
However, it is difficult and time consuming to generate a body fittinggrid for an interface problem in which the interface is complicatedgeometries or moves with time.
Moreover, due to the discontinuity of the coefficients along theinterface and the low global regularity of the solution, it is difficult toachieve higher order accuracy with standard finite element methods.
Therefore, few publications can be found on using body fitting gridsto solve moving interface problems with topological changes such asmerging and splitting.
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Unfitted finite element method
The first immersed finite element method was developed by Li [Appl.Numer. Math.1998] for solving the one-dimensional two-point boundaryvalue problem. Following this idea,
Zhilin Li et al. [Numer. Math., 2003; Numerical Methods for PDEs,2004] proposed an immersed finite element method using uniformCartesian triangular grids and their numerical examples demonstratedan optimal order of the errors.
So-Hsiang Chou [Adv. Comput. Math., 2010] derived optimal errorestimates of immerse finite element method for second order ellipticequations with discontinuous coefficients.
Jinru Chen and Zhilin Li [J. Sci. Comput., 2014; NumericalAlgorithms, 2016] proposed a symmetric and consistent, and a newaugmented immersed finite element method for interface problems.
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Wenqiang Feng, Xiaoming He, Yanping Lin, and Xu Zhang,[Communications in Computational Physics, 2014]proposed theIFE-AMG algorithm to solve the linear systems of the bilinear andlinear IFE methods for both stationary and moving interface problems.
Tao Lin, Yanping Lin and Xu zhang, [SIAM J. Numer. Amal., 2015,etc] studied DG immersed finite element methods.
Zhimin Zhang, [Adv. Comput. Math., 2017; IMA J. Numer. Anal.,2017] studied superconvergence properties.... ...
Other unfitted finite element methods
Jianguo Huang and Jun Zou, [IMA J. Numer. Anal., 2002] a mortarelement method.
Peter Hansbo et al., [Comput. Method Appl. M., 2002] An unfittedfinite element method, based on Nitsche’s method.
Haijun Wu and Yuanming Xiao, [arXiv:1007.2893] an unfittedhp-interface penalty finite element method.
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Semi-linear interface model
We consider the following semi-linear elliptic interface problems withdiscontinuous diffusion coefficients:
∇ · (β∇u) = f (x , u), x ∈ Ω. (1)
Let Ω be a convex polygonal domain in R2, Ω1 ⊂ Ω be an opendomain with C 2 boundary Γ = ∂Ω1 ⊂ Ω, and Ω2 = Ω\Ω1.
2
1
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The system is subjected to the boundary condition:
u = 0, on ∂Ω, (2)
and the jump conditions on the interface
[u] = 0,
[β∂u
∂n
]= 0 across Γ, (3)
where [v ] is the jump of a quantity v across the interface Γ and n the unitoutward normal to the boundary ∂Ω1. For ease of exposition, we assumethat the coefficient function β is positive and piecewise constant, i.e.
β(x) = β1 for x ∈ Ω1; β(x) = β2 for x ∈ Ω2. (4)
We will also write f (x , ξ) := f (ξ) and ∂f (x , ξ)/∂ξ := f ′(ξ) for simplicity.
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Weak formulation
The weak form for the interface problems (1)-(3) is stated as: findu ∈ H1
0 (Ω) such that
a(u, v) = (f (u), v), ∀v ∈ H10 (Ω), (5)
where
a(u, v) =
∫Ωβ∇u · ∇vdx , ∀u, v ∈ H1
0 (Ω).
For the analysis, we introduce the following space
H2(Ω) := u ∈ H1(Ω) : u ∈ H2(Ωs), s = 1, 2
equipped with the norm
‖u‖2eH2(Ω):= ‖u‖2
H2(Ω1)+ ‖u‖2
H2(Ω2), ∀u ∈ H2(Ω).
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Regularity theorem
Then we have the following regularity theorem for the weak solution uof the variational problem (5); see [Bramble and King, Adv. Comput.Math., 1996; Pierre Grisvard, SIAM, 2011].
Lemma 1.1
If u ∈ H10 (Ω) and f ∈ Lq(Ω), for 1 < q ≤ 2. Then the variational problem
(5) has a unique solution u ∈ H2(Ω), for some constant C > 0, whichsatisfies
‖u‖eH2(Ω)≤ C‖f ‖Lq(Ω). (6)
where the constant C depends on q, and the domain Ω.
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Some assumptions of f (u)
The weak assumption on the nonlinearity allows for a large class ofnonlinear problems containing both monotone and nonmonotonenonlinearities.
Assume f : Ω×R → R is a Caratheodory function, which satisfiesthe barrier-sign conditions in its second argument: there exist constantsρ, σ ∈ R, with ρ ≤ σ, such that
f (x , ξ) ≤ 0, ∀ξ ≥ σ, a.e. in Ω,
f (x , ξ) ≥ 0, ∀ξ ≤ ρ, a.e. in Ω.
This assumption is somewhat different but contains larger class ofnonlinear problems than [Sinha and Deka, Appl. Numer. Math., 2009],where their assumption is
|f ′(ξ)| ≤ C |ξ| and |f ′′(ξ)| ≤ C , ∀ξ ∈ R.
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Then we have the following priori L∞ bounds (see [Michael Holst etal., Numerical Methods for PDEs, 2013], Theorem 2.3).
Lemma 1.2
Let the assumption of f above hold. Let u ∈ H10 (Ω) be any weak solution
to (1). Then, we have
u1 ≤ u ≤ u2, a.e. in Ω, (7)
for the constants u1 and u2 defined by
u1 = minρ, 0, u2 = maxσ, 0.
The priori L∞ bounds play crucial roles in controlling the nonlinearityensuring that the nonlinearity f has a certain ”local Lipschitz” property.With this property, we can prove the error estimates later.
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1 Introduction
2 Immersed interface finite element space
3 Immersed interface finite element method for semi-linear interfaceproblems
4 Two-grid algorithms for semi-linear interface problems
5 Conclusions and future work
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Immersed interface finite element space
Let Th be the usual shape regular finite element triangulations withmesh size h that covers the domain Ω. Remark that in practice thecomputational domain Ω can often be chosen as a rectangular domainwith sides parallel to the coordinate axes, and the finite element mesh canbe uniform.
An element T ∈ Th is an interface element if the interface Γ passesthrough the interior of T , otherwise the element T is a non-interfaceelement. Let T ∗
h be the collection of all interface elements. Note that ifone of its edges is part of the interface, the element is a non-interfaceelement. Assume that the interface meets the edges of an interfaceelement at no more than two points.
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Triangular Cartesian meshes independent of theinterface
Interface element
Non interface element
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Immersed interface finite element space
An immersed interface finite element space Vh can be defined withrespect to the mesh Th to be a finitely dimensional subspace of L2(Ω) thatconsists of all the linear combinations of the corresponding basis functionsφ1, φ2, · · · , φN for some integer N ≥ 1:
Vh = spanφ1, φ2, · · · , φN. (8)
As usual, we want to construct local basis functions on each element T ofthe partition Th. For a non-interface element T ∈ Th, we simply use thestandard linear shape functions on T , and use Sh(T ) to denote the spacesspanned by the three nodal basis functions on T .
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Interface element
Consider a typical interface element T whose geometric configuration isgiven in the following Figure in which three vertices are given by A = (0, h),B = (0, 0), C = (h, 0), and the curve between points D and E is a part of theinterface across which quantity β has a jump. Let DE be the line segmentconnecting the intersections of the interface and the edges of a triangle T .Assume that the coordinates at D and E are (0, y1) and (h − y2, y2), respectively.The line segment divides T into two parts T1 and T2, one triangular and theother quadrilateral. The local basis function for a general interface element in themesh Th can be defined through the usual affine transformation.
(0, )A h
( ,0)C hB
(0,0)
D
E1(0, )y
2 2( , )h y y
1T
2T2
1
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Local basis functions on interface element
Each of the three local basis functions corresponding to the nodes A,B, or C takes value 1 at one node and 0 at the other two. Once the valuesat nodes A, B, and C are specified, a local nonconforming finite elementbasis function ϕ for this interface triangle is determined by
ϕ(x , y) =
ϕ1(x , y) = a0 + a1x + a2(y − h) in T1,
ϕ2(x , y) = b0 + b1x + b2y in T2,(9)
The coefficients ai and bi (i = 0, 1, 2) are determined by the conditions
ϕ1(D) = ϕ2(D), ϕ1(E ) = ϕ2(E ), β1∂ϕ1
∂n= β2
∂ϕ2
∂n, (10)
where n is the unit normal direction of the segment DE .
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Theorem 2.1 (Li et al., Numer. Math., 2003; Chou et al., Adv.Comput. Math., 2010)
Given a right triangle ABC as indicated in the above figure. The piecewiselinear function ϕ(x , y) defined by (9) and (10) is uniquely determined byϕ(A), ϕ(B), ϕ(C ).
Remark 2.1
Note that basis functions defined in this way can be discontinuous acrossedges of interface elements. So, this defines a nonconforming finiteelement. We note that the immersed finite element space Vh is amodification to the standard piecewise linear conforming finite elementspace when the coefficient β is discontinuous; the two spaces are the samewhen β1 = β2.
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We need introduce the following spaces. For any T ⊂ Ω,
fW m,p(T ) = u : u|Ts ∈ W m,p(Ts), s = 1, 2, p ≥ 1, m ≥ 0,
eH2int(T ) =
nu ∈ H1(T ) : u|Ts ∈ H2(Ts), s = 1, 2,
»β
∂u
∂n
–= 0 on Γ ∩ T
oand for any u ∈ W m,p(T ),
‖u‖2m,p,T = ‖u‖2
m,p,T1+ ‖u‖2
m,p,T2, |u|2m,p,T = |u|2m,p,T1
+ |u|2m,p,T2,
where ‖ · ‖m,p,Ts is the norm of W m,p(Ts), s = 1, 2. When p = 2, we
define Hm(T ) = W m,2(T ) as usual and denote its norm by ‖ · ‖m,T .Furthermore, we define H1/2(e) as the trace space on an edge e of T
of all functions in H1(T ) with the norm
‖v‖1/2,e = infu∈H1(T )
u|e=v
‖u‖1,T (11)
and H−1/2(e) as the dual space of H1/2(e), where the norm is given by
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‖u‖−1/2,e = supv∈H1/2(e)
〈u, v〉e‖v‖1/2,e
, (12)
where 〈·, ·〉 is the duality pairing.Next, we introduce a transfer operator γ : Vh(T ) → Sh(T ). Define
two trace spaces on an edge e of T :
Vh(e) = φ|e : φ ∈ Vh(T ), Sh(e) = φ|e : φ ∈ Sh(T ). (13)
Now we define γe : Vh(e) → Sh(e) by γe φ|e := (γφ)|e . We introduce thefollowing estimate [Chou et al., Adv. Comput. Math., 2010].
Lemma 2.1
Let T be an interface element and e an edge of T . Then the followinginequality holds for all φ ∈ Vh(T ):∥∥∥φ|e − γe φ|e
∥∥∥−1/2,e
≤ Cγh|φ|1,T , (14)
where Cγ is a constant independent of h and interface points.
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Interpolation operator
For any u ∈ H2int(T ), let Πhu ∈ Vh. The interpolation Πh can be
naturally extended such that Πh: H2int(Ω) → Vh(Ω).
Then, an estimate of the interpolation was given in the followingtheorem; see [Li et al., Numer. Math., 2003].
Theorem 2.2
Let T be an interface element. Then there exists a constant C > 0 suchthat
‖u − Πhu‖m,T ≤ Ch2−m‖u‖2,T , m = 0, 1, (15)
for any u ∈ H2int(T ).
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1 Introduction
2 Immersed interface finite element space
3 Immersed interface finite element method for semi-linear interfaceproblems
4 Two-grid algorithms for semi-linear interface problems
5 Conclusions and future work
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Immersed finite element method
We now consider the immersed interface finite element method forsemi-linear interface problems: find uh ∈ Vh(Ω) such that
ah(uh, φ) = (f (uh), φ), ∀φ ∈ Vh, (16)
where
ah(u, v) =∑T∈Th
∫Tβ∇u · ∇vdx , ∀u, v ∈ Hh(Ω), (17)
where Hh(Ω) := v | v |T ∈ H1(T ), ∀T ∈ Th and Hh(Ω) is endowed withthe broken H1 semi-norm as ‖v‖2
1,h :=∑
T∈Th|v |21,T .
Remark 3.1
Note that the bilinear operator ah(·, ·) is bounded and coercivity (see [Li etal., Numer. Math., 2003; Chou et al., Adv. Comput. Math., 2010]).
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For the energy norm error estimate of the immersed interface finiteelement method for semi-linear elliptic interface problems, we can provethe following lemma similar to the well-known second Strang lemma.
Lemma 3.1
Let u ∈ H2(Ω) and uh ∈ Vh be the solutions of the problem (5) and thefinite element equations (16), respectively. Then there exists a constantC > 0 such that
‖u − uh‖1,h ≤ C
inf
vh∈Vh(Ω)‖u − vh‖1,h
+ supwh∈Vh(Ω)
|ah(u,wh)− (f (uh),wh)|‖wh‖1,h
,
(18)
where C depends on α and M.
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Proof. By the coercivity of ah(·, ·),
α‖uh − vh‖21,h
≤ah(uh − vh, uh − vh)
=ah(u − uh, vh − uh) + ah(vh − u, vh − uh)
≤ah(u, vh − uh)− (f (uh), vh − uh) + M‖u − vh‖1,h‖vh − uh‖1,h,
where α is the coercivity constant.Next, we deduce the left-hand side bound
ah(u − uh,wh) ≤ M‖u − uh‖1,h‖wh‖1,h, ∀wh ∈ Vh.
Thus,
‖u − uh‖1,h ≥ M−1 supwh∈Vh
ah(u,wh)− (f (uh),wh)
‖wh‖1,h.
where M is boundness constant of ah(·, ·).
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We now use the above lemma to prove the following broken H1-normerror estimate.
Theorem 3.1
Let u ∈ H2(Ω) and uh ∈ Vh be the solutions of the problem (5) and thefinite element equations (16), respectively. Then we have
‖u − uh‖1,h ≤ Ch‖u‖eH2(Ω). (19)
where C = max‖f ‖1,∞,Cγ is independent of h and the location of theinterface.
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Sketch of proof
Proof. The first term in (18) is trivial through Theorem 2.2:
infvh∈Vh(Ω)
‖u − vh‖1,h ≤ Ch‖u‖eH2(Ω). (20)
For the second term in (18), we have
ah(u,wh)− (f (uh),wh)
=∑T∈Th
∫Tβ∇u · ∇whdx −
∫Ω
f (uh)whdx
=(f (u)− f (uh),wh) +∑T∈Th
〈β∂u
∂n,wh〉∂T
=(I )1 + (I )2, (21)
where n is the unit outward normal vector on each ∂T , and wh ∈ Vh(Ω).
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For (I )1, we have used ‖u − Πhu‖L2(T ), and note that uh is apiecewise linear function in element T .
As for (I )2, we have∑T∈Th
〈β∂u
∂n,wh〉∂T =
∑T∈T ∗
h
〈β∂u
∂n,wh − γewh〉∂T
=∑
T∈T ∗h
∑e⊂∂T
〈β∂u
∂n,wh − γewh〉e
=∑
T∈T ∗h
∑e⊂∂T
2∑s=1
〈β∂u
∂n,wh − γewh〉es
where es = e ∩ Ωs , s = 1, 2 and note that∑T∈Th
〈β∂u
∂n, γewh〉∂T = 0.
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From Lemma 2.1, it follows that
〈β∂u
∂n,wh − γewh〉es ≤
∥∥∥∥β∂u
∂n
∥∥∥∥1/2,es
‖wh − γewh‖−1/2,es
≤Cγh‖u‖eH2(Ω)|wh|1,T . (22)
Thus summing over all the elements T ∈ Th, we get∑T∈Th
〈β∂u
∂n,wh〉∂T ≤ Ch‖u‖eH2(Ω)
|wh|1,h. (23)
Now the desired estimate (19) follows from (18), (20), (21) and (23).This finishes the proof of Lemma 3.1.
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Auxiliary problem
We now apply the duality argument to obtain Lp (2 ≤ p <∞) normerror estimates of the immersed finite element method for semi-linearinterface problems.
Let ω ∈ H2(Ω) be the solution of the following auxiliary problem: findω ∈ H2(Ω) satisfying
−∇ · (β∇ω)− f ′(u)ω = u − uh, in Ω,
w = 0, on ∂Ω,(24)
with jump conditions [ω] = 0 and[β ∂ω
∂n
]= 0 across the interface Γ. Given
2 ≤ p <∞, set q = p/(p − 1) ∈ (1, 2]. By Lemma 1.1, we know that
‖ω‖eH2(Ω)≤ C‖u − uh‖Lq(Ω). (25)
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Along with (24), let us also introduce the immersed interface finiteelement approximation: find ωh ∈ Vh satisfying
ah(ωh, vh)− (f ′(u)ωh, vh) = (u − uh, vh), ∀vh ∈ Hh. (26)
Now noting the fact the jump [β ∂ω∂n ] = 0 across Γ, and by immersed
finite element approximation theory for the linear interface problem(24)-(26) (cf. [Chou, Adv. Comput. Math., 2010]), we have
‖ω − ωh‖1,h ≤ C1h‖ω‖eH2(Ω)≤ Ch‖u − uh‖Lq(Ω). (27)
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Lp error estimate
The main results of this section are stated in the following theorem:
Theorem 3.2
Let u ∈ H2(Ω) and uh ∈ Vh be the solutions of the problem (5) and thefinite element equations (16), respectively. For 2 ≤ p <∞, we have
‖u − uh‖Lp(Ω) ≤ Ch2‖u‖eH2(Ω). (28)
where the constant C is positive and depends on f , u1, u2 and Cγ .
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Sketch of proof
Proof. Multiply vh = u − uh ∈ Hh(Ω) to both sides of (24) and by using(26), (5) and (16), we get
(u − uh, u − uh)
=∑T∈Th
∫Tβ∇(u − uh) · ∇ωdx − (f ′(u)ω, u − uh)
−∑T∈Th
∫∂T
(u − uh)β∂ω
∂nds
= ah(u − uh, ω − ωh)− (f ′(u)(ω − ωh), u − uh)− (f ′(u)(u − uh), ωh)
+ (f (u)− f (uh), ωh) +∑T∈Th
∫∂Tβ∂u
∂nωhds −
∑T∈Th
∫∂T
(u − uh)β∂ω
∂nds
= : (II )1 + (II )2 + (II )3 + (II )4 + (II )5 + (II )6. (29)
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We need use ‖u − uh‖1,h, ‖ω − ωh‖1,h error estimate for (II )1, (II2).By Taylor expansion, we have
(II )3 + (II )4 =(f (u)− f (uh)− f ′(u)(u − uh), ωh)
=− (1
2f ′′(ξ1)(u − uh)
2, ωh), (30)
where ξ1 is some function value.Therefore, by Holder inequality, the (II )3 + (II )4 term in (29) can be
bounded as:
|(II )3 + (II )4| ≤‖f ‖2,∞‖u − uh‖20,2m‖ωh‖0,m/(m−1)
≤Ch2‖u‖eH2(Ω)‖u − uh‖0,q, (31)
where we choose m > 2p/(p + 2).
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As for (II )5, we introduce the extended transfer operator γ:Vh(T )⊕ spanω → Sh(T )⊕ spanω defined by
γ(φ+ cω) := γφ+ cω, for φ ∈ Vh, c ∈ R.
Define γe on each edge of T by
γe(φ+ cω)|e := (γφ+ cω)|e .
It is clear that ‖γe‖ is bounded above by a constant independent of h andthe location of the interface for any norm ‖ · ‖. Now applying an analysissimilar to obtain (23), we rewrite (II )5 as
(II )5 =∑T∈Th
∫∂Tβ∂u
∂n(ωh − ω)ds
=∑T∈Th
∑e⊂∂T
∫eβ∂u
∂n[(ωh − ω)− γe(ωh − ω)]ds (32)
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Hence by Lemma 2.1, we get
|(II )5| ≤Ch‖u‖eH2(Ω)‖ω − ωh‖1,h ≤ Ch2‖u‖eH2(Ω)
‖ω‖eH2(Ω)
≤Ch2‖u‖eH2(Ω)‖u − uh‖Lq(Ω). (33)
Arguing as deriving (32), we can deduce (II )6 by interchanging therole of ω and u,
|(II )6| ≤Ch‖ω‖eH2(Ω)‖u − uh‖1,h
≤Ch2‖ω‖eH2(Ω)‖u − uh‖Lq(Ω), (34)
where we have used (27) in the last step.Now the Lp-error estimate (28) follows from the error estimates of
(II )1, (II )2, (II )3, (II )4, (II )5 and (II )6.
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1 Introduction
2 Immersed interface finite element space
3 Immersed interface finite element method for semi-linear interfaceproblems
4 Two-grid algorithms for semi-linear interface problems
5 Conclusions and future work
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Two-grid Methods
Immersed finite element approximation of semi-linear problems resultsin the need to solve systems of nonlinear algebraic equations, and thenumbers of unknowns in these systems can be extraodinary large. So, it isnecessary to study highly efficient and highly accurate algorithms fornon-linear systems.
The two-grid finite element method based on two finite elementspaces on one coarse and one fine grid was first introduced by Xu in 1994,1996 for the nonsymmetric and nonlinear elliptic problems.
J. Xu, A novel two-grid method for semilinear equations, SIAMJournal on Scientific Computing, 15:231-237, 1994.
J. Xu, Two-grid discretization techniques for linear and non-linearPDEs, SIAM Journal on Numerical Analysis, 33:1759-1777, 1996.
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As almost same time, we have proposed a multilevel iterative methodthat not only reduces the computing work but also preserves all of the highaccuracy properties such as superconvergence, extrapolation, etc, for finiteelement solutions to singular problems.
Y. Chen and Y. Huang, A multilevel iterative method for finiteelement solutions to singular two-point boundary value problems,Natural Science J. of Xiangtan University, 1994.
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With the development of this highly efficient method, the two-gridmethod was further investigated by many author. For instance,
C. N. Dawson, M. F. Wheeler and C. S. Woodward, nonlinearparabolic equations, SIAM J. Numer. Anal., 1998;
J. Xu and A. Zhou, eigenvalue problems, Math. Comp., 2001;
Yinnian He, Navier-Stokes equations , SIAM J. Numer. Anal., 2003;
Y. Chen, nonlinear reaction-diffusion problems, Int. J. Numer. Met.Eng., 2003, 2007; miscible displacement problem in a porous media,Commun. Comput. Phys., Sci. Comput., 2016; compressible miscibledisplacement problem, J. Comput. Appl. Math., 2018;
Jicheng Jin, nonlinear schrodinger equation, Math. Comput., 2006;Journal of Computational Mathematics, 2015;
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M. Mu and J. Xu, coupling fluid flow with porous media flow, SIAMJournal on Numerical Analysis, 2007.... ...
Now, however, there are few results about two-grid methods forsemi-linear interface problems by finite element methods (or immersedfinite element methods). Only one work by Michael Holst using fittedfinite element method:
M. Holst, R. Szypowski, and Y. Zhu, Two grid methods for semilinearinterface problems, Numerical Methods for Partial DifferentialEquations, 2013.
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The main idea of two-grid method is to reduce the solution of a
semi-linear elliptic problem on a given fine grid with mesh size h to the
solution of the same elliptic problem on a much coarser grid with mesh
size h H, which can be easily solved as the size of the discrete elliptic
problem is significantly smaller than the original elliptic problem on the
fine grid, and the solution of a linear problem on the same fine grid, which
can be solved by mature and efficient numerical algorithms.
Next, we will present the two-grid algorithms and analyze the
convergence accuracy. The fundamental ingredient in these schemes is
another immersed finite element space VH ⊂ Vh (h H < 1), defined on
a coarser uniform triangulation of Ω.
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To estimate our two-grid methods, we need the help of a localmonotonicity assumption on the nonlinearity. This approach was used in[Long Chen, M. Holst, and J. Xu, SIAM J. Numer. Anal., 2007; M. Holstet al., Commun. Comput. Phys., 2012] for analyzing the convergence ofadaptive methods for the Poisson-Boltzmann equation.
Assume f is locally monotone, namely,
f ′(ξ) ≤ 0, ∀ξ ∈ [u1, u2].
where u1 and u2 are the barriers defined in (7).For simplicity, let us denote
N(u; v , ϕ) =(f (u) + f ′(u)(v − u), ϕ
). (35)
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Two-grid method: Algorithm 1
Now, we present the two-grid method which has two steps as follows.
Algorithm 1
Step 1: On the coarse grid TH , we solve the nonlinear system (16) tocompute uH ∈ VH ,
aH(uH , vH) = (f (uH), vH), ∀vH ∈ VH . (36)
Step 2: On the fine grid Th, we compute Uh ∈ Vh to satisfy the followinglinear system:
ah(Uh, vh) = N(uH ;Uh, vh), ∀vh ∈ Vh. (37)
Algorithm 1 solves the original semi-linear interface problem on thecoarse grid Th, and then performs one Newton iteration on the fine grid.
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The remainder term of Algorithm 1
Fix any ξh ∈ Vh, let
η(t) =: ah(ψ(t), ξh)− (f (ψ(t)), ξh), (38)
with ψ(t) = uH + t(uh − uH).Then, by Taylor expansion and (16), we have
0 =ah(uh, ξh)− (f (uh), ξh) = η(1)
=η(0) + η′(0) +
∫ 1
0η′′(t)(1− t)dt
=ah(uH , ξh)− (f (uH), ξh) + ah(uh − uH , ξh)
− (f ′(uH)(uh − uH), vh) +
∫ 1
0η′′(t)(1− t)dt. (39)
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By direct calculation, the remainder term, denoted by R(uH , uh, ξh),has the following form:
R(uH , uh, ξh) = :
∫ 1
0η′′(t)(1− t)dt
=−∫ 1
0
(f ′′(ψ(t))(uh − uH)2, ξh
)(1− t)dt.
Therefore, we have the following estimate
|R(uH , uh, ξh)| ≤ CR‖uh − uH‖20,2p1
‖ξh‖0,q1 , (40)
for any p1, q1 ≥ 1 with 1/p1 + 1/q1 = 1, where CR depends on f , u1 andu2.
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Lemma 4.1
Let uh ∈ Vh be the solutions to (16) on Th and Uh ∈ Vh be theapproximated solution obtained by Algorithm 1. Then we have thefollowing estimate
‖uh − Uh‖1,h ≤ C (h4 + H4)‖u‖eH2(Ω). (41)
for some positive constant C depending on CR , f , u1 and u2.
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Proof. From Algorithm 1, we have
ah(uh − Uh, ξh)− (f ′(uH)(uh − Uh), ξh)
=ah(uh, ξh)− (f (uH), ξh)− (f ′(uH)(uh − uH), ξh)
=− R(uH , uh, ξh), ∀ξh ∈ Vh. (42)
Then, taking ξh = uh − Uh in (42), we obtain that
‖uh − Uh‖21,h .ah(uh − Uh, uh − Uh)
=(f ′(uH)(uh − Uh), uh − Uh)− R(uH , uh, uh − Uh)
≤− R(uH , uh, uh − Uh)
≤CR‖uh − uH‖20,2p1
‖uh − Uh‖1,h, (43)
where in the third step we used the assumption of f , and in the last stepwe used (40).
Then, Lemma 4.1 follows immediately from the error estimate‖u − uH‖Lp , ‖u − uh‖Lp and the triangle inequality.
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A priori error estimate of Algorithm 1
Then, we have the following theorem.
Theorem 4.1
Let u ∈ H10 be the solution of (5), and Uh ∈ Vh be the solution of
Algorithm 1. Then, We have
‖u − Uh‖1,h ≤ C (h + H4)‖u‖eH2(Ω). (44)
for some positive constant C.
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Sketch of proof
Proof. Since the immersed finite element space Vh is nonconforming, byusing Lemma 3.1, we have
‖u − Uh‖1,h ≤ C
inf
vh∈Vh(Ω)‖u − vh‖1,h
+ supϕ∈Vh(Ω)
|ah(u, ϕ)− N(uH ;Uh, ϕ)|‖ϕ‖1,h
.
(45)
From (16), (35) and Green’s formula, we have
ah(u, ϕ)− N(uH ;Uh, ϕ)
=∑T∈Th
∫Tβ∇u · ∇ϕdx −
∫Ω(f (uH) + f ′(uH)(Uh − uH))ϕdx
=(f (u)− f (uH)− f ′(uH)(Uh − uH), ϕ) +∑T∈Th
〈β∂u
∂n, ϕ〉∂T . (46)
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By Taylor expansion,
(f (u)− f (uH)− f ′(uH)(Uh − uH), ϕ)
=(f (u)− f (uh), ϕ) + (1
2f ′′(ξ2)(uh − uH)2, ϕ)
+ (f ′(uH)(uh − Uh), ϕ)
= : (III )1 + (III )2 + (III )3 (47)
where ξ2 is some function value between uh and uH . We need use theerror estimates ‖u − uH‖L4 , ‖u − uh‖Lp and ‖uh − Uh‖1,h for (III )1, (III )2and (III )3.
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For the last term in (46), arguing as deriving (23), we can deduce
∑T∈Th
〈β∂u
∂n, ϕ〉∂T =
∑T∈T ∗
h
∑e⊂∂T
2∑s=1
〈β∂u
∂n, ϕ− γeϕ〉es
≤∑
T∈T ∗h
∑e⊂∂T
2∑s=1
∥∥∥∥β∂u
∂n
∥∥∥∥1/2,es
‖ϕ− γeϕ‖−1/2,es
≤Ch‖u‖eH2(Ω)‖ϕ‖1,h. (48)
where es = e ∩ Ωs , s = 1, 2.Then, Theorem 4.1 follows immediately from (45), (20), (48) and the
previous estimates for (III )1, (III )2 and (III )3. This finishes the proof ofTheorem 4.1.
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Remark 4.1
According to Theorem 4.1, in order to obtain the optimal (or nearlyoptimal) approximation for the discretization Uh in H1 norm, it suffices totake H = h1/4. To get an idea numerically if the fine mesh size h = 2−24
gives dim Vh ≈ 2.8× 1014, the coarse mesh size H could be H = 1/64which gives dim VH ≈ 3.9× 103.
Algorithm 1 presented above can be greatly improved if one furthercorrection step is carried out on Vh. The third correction step (whichneeds very little extra work) improves the accuracy of Algorithm 1. Wehave the following three step two-grid algorithm.
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Two-grid method: Algorithm 2
Algorithm 2
Step 1: On the coarse grid TH , we solve the nonlinear system (16) to computeuH ∈ VH ,
aH(uH , vH) = (f (uH), vH), ∀vH ∈ VH . (49)
Step 2: On the fine grid Th, we compute Uh ∈ Vh to satisfy the following linearsystem:
ah(Uh, vh) = N(uH ;Uh, vh), ∀vh ∈ Vh. (50)
Step 3: On the fine grid Th, solving the following linear system for eh ∈ Vh:
ah(eh, vh) = (f ′(uH)eh +1
2f ′′(uH)(Uh − uH)2, vh), ∀vh ∈ Vh. (51)
Set uh = Uh + eh.
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We define the same remainder term as Algorithm 1. We have thefollowing error estimate:
Lemma 4.2
Let uh ∈ Vh be the solutions of (16) on Th and uh ∈ Vh be theapproximated solution obtained by Algorithm 2. Then we have thefollowing estimate
‖uh − uh‖1,h ≤ C (h4 + H4)‖u‖eH2(Ω). (52)
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Proof. From Algorithm 2, we have
ah(uh − uh, ζh)− (f ′(uH)(uh − uh), ζh)− (1
2f ′′(uH)(Uh − uH)2, ζh)
=ah(uh, ζh)− (f (uH), ζh)− (f ′(uH)(uh − uH), ζh)
=− R(uH , uh, ζh), ∀ζh ∈ Vh. (53)
Then, taking ζh = uh − uh, we obtain that
‖uh − uh‖21,h .ah(uh − uh, uh − uh)
=(f ′(uH)(uh − uh), uh − uh) + (1
2f ′′(uH)(Uh − uH)2, ζh)
− R(uH , uh, uh − uh)
≤C‖f ‖2,∞‖Uh − uH‖2L4‖uh − uh‖1,h + C‖uh − uH‖2
L4(Ω)‖uh − uh‖1,h.
We need use the error estimates of ‖u −Uh‖1,h, ‖u − uH‖L4(Ω) and ‖u − uh‖L4(Ω).
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A priori error estimate of Algorithm 2
Thus, we have the following Theorem.
Theorem 4.2
Let u ∈ H2(Ω) be the solution of (5), and uh ∈ Vh be the solution ofAlgorithm 2. We have the following estimate
‖u − uh‖1,h ≤ C (h + H6). (54)
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Sketch of proof
Proof. Following Lemma 3.1, we have
‖u − uh‖1,h ≤ C
inf
vh∈Vh(Ω)‖u − vh‖1,h
+ supζh∈Vh(Ω)
|ah(u, ζh)− N(uH ; uh, ζh)− (12 f ′′(uH)(Uh − uH)2, ζh)|
‖ζh‖1,h
.
(55)
By (18), (17), (50) and (51), we have
ah(u, ζh)− N(uH ; uh, ζh)− (1
2f ′′(uH)(Uh − uH)2, ζh)
=∑T∈Th
∫Tβ∇u · ∇ζhdx −
∫Ω(f (uH) + f ′(uH)(uh − uH)
+1
2f ′′(uH)(Uh − uH)2)ζhdx
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=(f (u)− f (uH)− f ′(uH)(uh − uH)
− 1
2f ′′(uH)(Uh − uH)2, ζh) +
∑T∈Th
〈β∂u
∂n, ζh〉∂T . (56)
By Taylor expansion,
(f (u)− f (uH)− f ′(uH)(uh − uH)− 1
2f ′′(uH)(Uh − uH)2, ζh)
=(f (u)− f (uh), ζh) + (1
2f ′′(uH)((uh − uH)2 − (Uh − uH)2), ζh)
+ (1
6f ′′′(ξ4)(uh − uH)3, ζh)
= : (IV )1 + (IV )2 + (IV )3 (57)
where ξ4 is some function value between uh and uH .
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We use the error estimate ‖u − uH‖L6 , ‖u − uh‖L6 , ‖u − uH‖L4 ,‖u − Uh‖1,h and ‖ζh − γeζh‖−1/2,e for (IV )1, (IV )2, (IV )3 and the lastterm in (56).
Remark 4.2
According to Theorem 4.2, in order to obtain the optimal (or nearlyoptimal) approximation for the discretization uh in H1 norm, it suffices totake H = h1/6. To get an idea numerically if the fine mesh size h = 2−24
gives dim Vh ≈ 2.8× 1014, the coarse mesh size H could be H = 1/16which gives dim VH ≈ 225.
Algorithm 1 presented above can be greatly improved if one furtherNewton iteration is carried out on Vh. We have the following three steptwo-grid algorithm.
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Two-grid method: Algorithm 3
Algorithm 3
Step 1: On the coarse grid TH , we solve the nonlinear system (16) to computeuH ∈ VH ,
aH(uH , vH) = (f (uH), vH), ∀vH ∈ VH . (58)
Step 2: On the fine grid Th, we compute Uh ∈ Vh to satisfy the following linearsystem:
ah(Uh, vh) = N(uH ;Uh, vh), ∀vh ∈ Vh. (59)
Step 3: On the fine grid Th, solving the following linear system for Uh ∈ Vh:
ah(Uh, vh) = N(Uh;Uh, vh), ∀vh ∈ Vh. (60)
Algorithm 3, roughly speaking, is to use the solution of Algorithm 1 as an
initial guess for one Newton iteration on the fine grid.
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The remainder term of Algorithm 3
Fix any χh ∈ Vh, let
η1(t) =: ah(ψ1(t), χh)− (f (ψ1(t)), χh), (61)
with ψ1(t) = Uh + t(uh − Uh). Then, by Taylor expansion and (16), wehave
0 =ah(uh, χh)− (f (uh), χh) = η1(1)
=η1(0) + η′1(0) +
∫ 1
0η′′1(t)(1− t)dt
=ah(Uh, χh)− (f (Uh), χh) + ah(uh − Uh, χh)
− (f ′(Uh)(uh − Uh), χh) +
∫ 1
0η′′1(t)(1− t)dt. (62)
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By direct calculation, the remainder term, denoted by R1(Uh, uh, χh),has the following form:
R1(Uh, uh, χh) :=
∫ 1
0η′′1(t)(1− t)dt
=−∫ 1
0(f ′′(ψ1(t))(uh − Uh)
2, χh)(1− t)dt.
Therefore, we have the following estimate
|R1(Uh, uh, χh)| ≤ CR1‖uh − Uh‖20,2p2
‖χh‖0,q2 , (63)
for any p2, q2 ≥ 1 with 1/p2 + 1/q2 = 1.
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Lemma 4.3
Let uh ∈ Vh be the solutions to (16) on Th and Uh ∈ Vh be theapproximated solution obtained by Algorithm 3. Then we have thefollowing estimate
‖uh − Uh‖1,h ≤ C (h8 + H8)‖u‖eH2(Ω). (64)
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Proof. From Algorithm 3, we have
ah(uh − Uh, χh)− (f ′(Uh)(uh − Uh), χh)
=ah(uh, χh)− (f (Uh), χh)− (f ′(Uh)(uh − Uh), χh)
=− R1(uH , uh, χh), ∀χh ∈ Vh. (65)
Then, taking χh = uh − Uh in (65), we obtain that
‖uh − Uh‖21,h .ah(uh − Uh, uh − Uh)
=(f ′(uH)(uh − Uh), uh − Uh)− R1(uH , uh, uh − Uh)
≤C‖uh − Uh‖21,h‖uh − Uh‖1,h, (66)
where the positive constant C depends on CR1 and the Poincare constant.Then, Lemma 4.3 follows immediately from the error estimate of
‖uh − Uh‖1,h. This completes the proof.
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A priori error estimate of Algorithm 3
Thus, we have the following Theorem.
Theorem 4.3
Let u ∈ H2(Ω) be the solution of (5), and Uh ∈ Vh be the solution ofAlgorithm 3. We have the following estimate
‖u − Uh‖1,h ≤ C (h + H8)‖u‖eH2(Ω). (67)
for some positive constant C depending on f , u1, u2, CR1 and Cγ .
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Sketch of proof
Proof. Following Lemma 3.1, we have
‖u − Uh‖1,h ≤ C
inf
vh∈Vh(Ω)‖u − vh‖1,h
+ supχh∈Vh(Ω)
|ah(u, ϕ)− N(Uh;Uh, χh)|‖χh‖1,h
.
(68)
By (18), (17) and (60), we have
ah(u, χh)− N(Uh;Uh, χh)
=∑T∈Th
∫Tβ∇u · ∇χhdx −
∫Ω(f (Uh) + f ′(Uh)(Uh − Uh))χhdx
=(f (u)− f (Uh)− f ′(Uh)(Uh − Uh), χh) +∑T∈Th
〈β∂u
∂n, χh〉∂T . (69)
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By Taylor expansion,
(f (u)− f (Uh)− f ′(Uh)(Uh − uH), χh)
=(f (u)− f (uh), χh) + (1
2f ′′(ξ3)(uh − Uh)
2, χh)
+ (f ′(uH)(uh − Uh), χh)
= : (V )1 + (V )2 + (V )3, (70)
where ξ3 is some function value between uh and Uh.We need use error estimates ‖u − uh‖L2(Ω), ‖u −Uh‖1,h, ‖uh −Uh‖1,h
and ‖χh − γeχh‖−1/2,e for (V )1, (V )2, (V )3 and the last term in (69).
Remark 4.3
According to Theorem 4.3, in order to obtain the optimal (or nearlyoptimal) approximation for the discretization Uh in H1 norm, it suffices totake H = h1/8. To get an idea numerically if the fine mesh size h = 2−24
gives dim Vh ≈ 2.8× 1014, the coarse mesh size H could be H = 1/8which gives dim VH ≈ 49.
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1 Introduction
2 Immersed interface finite element space
3 Immersed interface finite element method for semi-linear interfaceproblems
4 Two-grid algorithms for semi-linear interface problems
5 Conclusions and future work
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Conclusions and future work
Conclusions
The immersed interface finite element method is used to semi-linearinterface problems, and the optimal error estimates in broken H1 andLp norm are derived.
We propose three two-grid methods of the immersed interface finiteelement solutions for semi-linear interface problems and prove thepriori error estimates.
Future work
We will prove the Lp error estimates of the two-grid methods.
Numerical experiments
Two-grid methods for transient advection-diffusion equations withinterfaces.
Two-grid methods for semi-linear parabolic interface problems.
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Acknowledgements
Thank you!
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