CONVERGENCE FOR STABILISATION OF DEGENERATELY ...

CONVERGENCE FOR STABILISATION OFDEGENERATELY CONVEX MINIMISATION

PROBLEMS

S. BARTELS, C. CARSTENSEN, P. PLECHAC, AND A. PROHL

Abstract. Degenerate variational problems often result from arelaxation technique in effective numerical simulation of nonconvexminimisation problems. The relaxed energy density is the convexenvelope of the original one and so convex but not strictly convex.Hence strong convergence of straightforward finite element approx-imations cannot be expected but is relevant in many applications.This paper establishes a modified discretization by stabilisationand proves its convergence in strong norms.

1. Motivation and Introduction

The relaxation procedure in the calculus of variations allows the di-rect macroscopic simulation of models with finer and finer oscillations[L1, L2]. For the discrete problem that means that the non-convexenergy density is removed and replaced by some quasiconvex envelopeor —in some applications— even its convex envelope; we refer to theExample 1.1 for an illustration. The resulting discrete problem is thendegenerated in the sense that it is convex but not strictly convex andso the Newton solver faces situations where the Hessian matrix for thetangential stiffness matrix is not positive definite and may be singular.Standard numerical regularisations are analysed in this paper as stabil-isation techniques. Example 1.1 illustrates that the stabilisation allowsless Newton-iterations than the original relaxed problem. We prove forrelevant examples that proper stabilisation maintains the convergencerates of the discrete problem, and, which came much to a surprise,yields even strong convergence of the strain variables in certain cir-cumstances.

Key words and phrases. degenerate variational problems, convexification, stabil-isation, strong convergence, Euler-Lagrange equations, calculus of variations.

1

2 S. BARTELS, C. CARSTENSEN, P. PLECHAC, AND A. PROHL

Example 1.1 (3-Well Problem). Given Ω = (0, 1)2 and boundary datauD(x) = v0(x1) + v0(x2) for x = (x1, x2) ∈ Ω and

v0(t) =

(t− 1/4)3/6 + (t− 1/4)/8 for t ≤ 1/4,

−(t− 1/4)5/40− (t− 1/4)3/8 for t ≥ 1/4,

the relaxation W ∗∗ (i.e. the lower convex envelope) of the 3-well energydensity

W (F ) = min|F |2, |F − (1, 0)|2, |F − (0, 1)|2leads to the energy minimisation problem

minu∈A

E(u) for A = v ∈ W 1,2(Ω) : v = uD on ∂Ω and

E(u) =

∫Ω

W ∗∗(Du) dx+

∫Ω

|uD − u|2 dx+

∫Ω

fv dx

with f = divDW ∗∗(DuD). The exact solution of the relaxed min-imisation problem reads u(x) = uD(x) for x ∈ Ω. Its finite elementapproximation is computed on a sequence of uniform triangulationsT of Ω with mesh-size h = 1/2, 1/4, ..., 1/32 and degrees of freedomN = 1, 9, 49, 225, 961 into triangles which are translated copies ofconv(0, 0), (0, h), (h, h) and conv(0, 0), (h, h), (h, 0). Notice thatW ∗∗ vanishes identically in conv(0, 0), (1, 0), (0, 1) ⊂ R2 and hence sta-bilisation is in order. The resulting discrete problem reads

minuh∈Ah

Eh(uh) for Eh(uh) = E(uh) + hγ−1

∫Ω

|Duh|2 dx

and Ah = vh ∈ S1(T ) : vh = uD,h on ∂Ω where S1(T ) ⊆ W 1,2(Ω) isthe lowest order finite element space related to T and uD,h(z) = uD(z)for all nodes z on ∂Ω.

For the exponents γ = 0, 1/2, 1, 2 and γ = ∞ (γ = ∞means Eh = E,i.e. no stabilisation) we run a nested Newton-Raphson scheme. Thetermination criterion was an `2 norm of the residual less than 10−9.Table 1 displays the history of iteration numbers K as a function of γand h. This experimental result supports our interpretation from gen-eral observation that stabilisation improves the efficiency of the discreteproblem solve.

The paper is concerned with the convergence behaviour of the per-turbed discrete solutions. The class of problems analysed in this paperreads as follows. A natural finite element discretization of the Euler-Lagrange equations of a degenerately convex minimisation problem

(P ) Seek u ∈ A with

∫Ω

S(Du) : Dv dx+ J(u; v) = 0 for all v ∈ AD

STABILISATION OF DEGENERATE MINIMISATION PROBLEMS 3

h 1/2 1/4 1/8 1/16 1/32

γ = 0 4 4 5 7 8γ = 1/2 4 4 5 10 9γ = 1 4 4 5 13 16γ = 2 4 6 10 29 -γ = ∞ 4 10 98 - -

Table 1. Iteration numbers K required in the relaxed3-well problem of Example 1.1 as a function of uniformmesh-size h and parameter γ. A minus sign means noconvergence within 250 iteration steps.

(colon denotes the scalar product in Rm×n) with discrete spaces Ah =uD,h +AD,h and AD,h ⊆ AD reads

(Ph) Seek uh ∈ Ah with

∫Ω

S(Duh) : Dvh dx+ Jh(uh; vh) = 0

for all vh ∈ AD,h.

Typically, the nonlinear stress-strain function S : Rm×n → Rm×n isthe derivative S = Dϕ of an energy density function ϕ that is (quasi-)convex but not strictly (quasi-)convex. Lacking uniform convexity of ϕand so lacking uniform monotonicity of S we cannot generally expectstrong convergence of the error e := u− uh, namely

(1.1) limh→0

‖De‖Lp(Ω) = 0,

if an underlying mesh Th becomes finer and finer such that the maximalmeshsize tends to zero as h → 0. Instead of (1.1), one may merelyexpect weak convergence Duh Du in Lp(Ω) or convergence in weakernorms, e.g. limh→0 ‖u− uh‖Lr(Ω) = 0. It turns out that the continuouslower order term J : W 1,p(Ω; Rm) → W 1,p(Ω; Rm)∗ as well as boundaryconditions in A := v ∈ W 1,p(Ω; Rm) : v = uD on ΓD from some partΓD of the boundary ∂Ω of the domain Ω determine whether solutions uor uh are unique or not; we refer to Section 2 for detailed assumptions.A typical time-step in evolution of phase transitions leads to (P ) withan L2-uniformly convex low-order term J (see, e.g. [CP3]) and requiresstrong convergence of gradients.

It is the aim of this paper to introduce stabilisation strategies toguarantee (1.1). For a mesh-dependent bilinear form ah : Xh×Yh → Rsuch that Ah ⊆ Xh and AD,h ⊆ Yh we set Jh(uh, vh) := J(uh, vh) +


ah(uh, vh). For relaxed nonconvex minimisation problems the addi-tional term ah(uh, vh) allows a physical interpretation of a discrete sur-face energy. Provided that (P ) involves sufficient convexity, e.g. ifJ is uniformly monotone with respect to an Lp norm (on low-orderterms) and ϕ is convex, there exists a unique solution u of (P ). Then,if u ∈ H3/2+ε(Ω; Rm) for some ε > 0 we prove (1.1) for the uniquediscrete solution uh of (Ph).

In order to illustrate some of the arguments in the proof of (1.1)we avoid in this introduction any technicality through the (unrealistic)assumption Ah,AD,h ⊆ H2(Ω; Rm) and consider only one stabilisationterm

(1.2) Jh(uh; vh) := J(uh; vh) + h2

∫Ω

∆uh ·∆vh dx

(dot denotes the scalar product in Rm). Suppose furthermore that thelow-order term J is uniformly monotone such that standard argumentswith the Galerkin orthogonality yield

(1.3) h2‖∆e‖2L2(Ω) + ‖e‖2

L2(Ω) ≤ Ch2

for u ∈ H2(Ω; Rm) ∩ A. Then, an integration by parts and e = 0 on∂Ω lead to

‖De‖2L2(Ω) =

∫Ω

De : Dedx = −∫

Ω

e ·∆e dx.

Cauchy’s inequality, Young’s inequality in the resulting upper bound,and (1.3) in the final step prove

‖De‖2L2(Ω) ≤ ‖e‖L2(Ω)‖∆e‖L2(Ω) ≤

h

2‖∆e‖2

L2(Ω) +h−1

2‖e‖2

L2(Ω) ≤ Ch.

Hence there holds strong convergence of gradients (1.1) for p = 2 ifu ∈ H2(Ω; Rm). Since this argumentation requires C1 conforming finiteelements the practical use of stabilisation (1.2) is limited. Therefore,this paper establishes three discrete stabilisations which lead to (1.1)in case that Ah, AD,h are lowest order finite element spaces.

It should be stressed that stabilisation is in fact equivalent to [NW]stated for m = n = 1 and for a numerical modification that replacesJ by a lumped version Jh. The proof of (1.1) in [NW] employs par-ticular one-dimensional arguments for a particular model example. Incontrast to this, stabilisation as introduced in this paper, appears tobe a robust and flexible tool for a large class of degenerately convex


minimisation problems. Convergence rates for the gradient error, how-ever, requires strong regularity conditions of the exact solution alongwith its uniqueness.

The remaining part of this paper is organised as follows. The generalsetting and the main results are presented in Section 2. A list of ex-amples for S and J that meet the abstract framework in (P ) are givenin Section 3. In Section 4 we prove the main result. Notation and ba-sic results related to finite element discretizations are introduced andrecalled in Section 5. Sections 6-8 are devoted to three different sta-bilisations that define (Ph) and lead to (1.1) via the abstract result ofSection 2. Section 9 discusses strong convergence for a 2-well problemwhich results from a model for phase transitions in crystalline solids.Numerical examples are reported on in [Ba].

2. General Setting and Main Result

This section is devoted to a general framework that allows severalparticular choices of the stabilisation term ah for a large class of ex-amples indicated below. For this section, Jh is quite general and couldmodel a numerical quadrature for J as well.

Given a bounded Lipschitz domain Ω ⊂ Rn with polygonal (for n =2) or polyhedral (for n = 3) boundary ∂Ω and 1 < q ≤ 2 ≤ p < ∞,1/p+ 1/q = 1. Given uD ∈ W 1,p(Ω; Rm) set

AD := W 1,p0 (Ω; Rm) and A := uD +AD,

with W 1,p0 (Ω; Rm) = v ∈ W 1,p(Ω; Rm) : v|∂Ω = 0; | · |W 1,p(Ω) ab-

breviates the seminorm |v|W 1,p(Ω) := ‖Dv‖Lp(Ω) of v ∈ W 1,p(Ω; Rm).

For a discrete space AD,h ⊆ W 1,p0 (Ω; Rm), spaces Xh and Yh, and an

approximation uD,h of uD we merely suppose

Ah = uD,h +AD,h ⊆ Xh and AD,h ⊆ Yh.

The stress function S : Mm×n → R, the low-order terms

J : W 1,p(Ω; Rm) → (W 1,p(Ω; Rm))∗ and Jh : Xh → Y ∗h

of the continuous and discrete level, respectively, are supposed to sat-isfy the following hypotheses (H1)-(H3) for the exact and the discretesolution u ∈ A and uh ∈ Ah, respectively. A list of examples followsbelow in Section 3.


(H1). There exist positive constants α, r, s with 1 < r ≤ 2, 0 ≤ s <∞, and a function S : Mm×n → Mm×n such that, for all A,B ∈ Mm×n,

|S(A)− S(B)|r ≤ α(1 + |A|s + |B|s)(S(A)− S(B)) : (A−B).

Here, Mm×n denotes the real m× n matrices, | · | the Frobenius normrelated to the scalar product

A : B =m∑

k=1

n∑`=1

Ak `Bk ` for A,B in Mm×n.

(H2). There exist solutions u and uh of (P ) and (Ph), respectively.[Their uniqueness is not assumed explicitly, at this stage, any choicewill do it. However, the uniqueness of u is later an implication ofour strong regularity assumption.] That is suppose that u ∈ A withσ := S(Du) and uh ∈ Ah with σh := S(Duh) satisfy∫

Ω

σ : Dv dx+ J(u; v) = 0 for all v ∈ AD,∫Ω

σh : Dvh dx+ Jh(uh; vh) = 0 for all vh ∈ AD,h.

Throughout this paper, set

e := u− uh and δ := σ − σh.

(H3). There exist a constant B > 0, a strictly convex functionβ : [0,∞) → [0,∞) with β(0) = 0, and seminorms ‖ · ‖Xh

and ‖ · ‖Yh

on the function spaces Xh and Yh with Ah ⊆ Xh ⊆ W 1,p(Ω; Rm) andAD,h ⊆ Yh ⊆ W 1,p(Ω; Rm) such that e ∈ Xh, e−AD,h ⊆ Yh, and

β(‖e‖Xh

)≤ Jh(u; e)− Jh(uh; e),

Jh(u, v)− Jh(uh; v) ≤ B‖e‖Xh‖v‖Yh

for the exact and discrete solution u and uh with the error e = u− uh

from (H2), and v ∈ e−AD,h.

Theorem 2.1. Suppose (H1)-(H3) and let β∗ denote the dual func-tional to β, i.e. β∗(t) = supst−β(s) : s ≥ 0. Then, for all eh ∈ AD,h,there holds

(1− 1/r)

∫Ω

δ : Dedx+ (1/c1) ‖δ‖rLq(Ω) + β

(‖e‖Xh

)≤ c2|e− eh|r/(r−1)

W 1,p(Ω) + β∗(2B‖e− eh‖Yh

)+ 2

(Jh(u; eh)− J(u; eh)

).

The constants c1 and c2 depend on α, p, r, s, and upper bounds for|u|W 1,p(Ω) and |uh|W 1,p(Ω).


Remark 2.1. It follows from (H1) that 0 ≤ δ : De almost everywhereon Ω; hence all the terms on the left-hand side in the estimate of thetheorem are non-negative.

Remark 2.2. It is known that β∗ : [0,∞) → [0,∞) is a convex functionwith β∗(0) = 0. In particular, for β(t) = t2/2 one finds β∗(t) = t2/2.

Remark 2.3. The bounds of |u|W 1,p(Ω) and |uh|W 1,p(Ω) may follow fromfurther natural growth conditions on S, J , and Jh which we have notstated here.

Throughout this paper we consider Jh = J |Xh×Yh+ ah for a continu-

ous bilinear form ah : Xh × Yh → R. Then we can replace (H3) by thefollowing hypothesis.

(H4). Let 0 < m ≤M <∞ satisfy

m‖e‖2L2(Ω) ≤ J(u; e)− J(uh; e),

J(u; v)− J(uh; v) ≤M‖e‖L2(Ω)‖v‖L2(Ω)

for all v ∈ W 1,p(Ω; Rm).

Proposition 2.2. Suppose (H4), Ah ⊆ Xh ⊆ W 1,p(Ω; Rm) and AD,h ⊆Yh ⊆ W 1,p(Ω; Rm) are such that e ∈ Xh and e − AD,h ⊆ Yh. Assumethat ah : Xh×Yh → R is a continuous bilinear form, ‖ · ‖2

Xh= ‖ · ‖2

Yh=

‖ · ‖2L2(Ω) +ah(·, ·), and Jh = J |Xh×Yh

+ah. Then, there holds (H3) with

β(t) = min1,m t2 and B := max1,M.

Proof. This follows directly from the definitions of Jh, ‖·‖Xh, ‖·‖Yh

.

3. Examples

Example 3.1 (p-Laplacian). An energy minimisation of |Du|p/p leadsto the p-Laplacian problem with operator S(F ) = |F |p−2F and 2 ≤p < ∞. Since (e.g. by a combination of Lemma 2.1-2.3 in [CK]) forany distinct A,B ∈ Rn and α = 1 + max1, p− 22 there holds

|S(A)− S(B)|2

(S(A)− S(B)) : (A−B)≤ α(|A|p−2 + |B|p−2)

and (H1) is valid with r = 2, s = p − 2. See [CM, LB] for furtherresults.

Example 3.2 (Optimal Design). The relaxed model for an optimal de-sign problem derived in [GKR] leads to a minimisation problem withenergy density ϕ(F ) = ψ(|F |) and S(F ) = Dϕ(F ). Given positive


parameters 0 < t1 < t2 and 0 < µ2 < µ1 with t1µ1 = t2µ2, the C1

function ψ : [0,∞) → [0,∞) is defined by ψ(0) = 0 and

ψ′(t) =

µ1 t if 0 ≤ t ≤ t1,

t1µ1 = t2µ2 if t1 ≤ t ≤ t2,µ2 t if t2 ≤ t.

The function S(F ) satisfies (H1) with r = 2, s = 0, and α = µ1 [CP1];cf. also [F].

Example 3.3 (Scalar 2-Well Problem). Given distinct wells F1, F2 ∈Rn, F1 6= F2, the relaxed scalar 2-well problem leads to a convexifiedminimisation problem with energy density

ϕ(F ) = max|F −B|2 − |A|2, 02

+ 4(|A|2 |F −B|2 − [AT (F −B)]2)(3.1)

where A = (F2 − F1)/2 and B = (F1 + F2)/2. and satisfies (H1) withr = 2, s = 2, and α = 4 max2, |F1 − F2|2 [CP1, F]. This scalarproblem can be deduced from the Ericksen-James energy density in ananti-plane shear model; the version for n = 1, due to O. Bolza [Bo],serves as a master example in non-convex minimisation [Y].

Example 3.4 (Compatible Vectorial 2-Well Problem). Given two sym-metric matrices E1, E2 ∈ Mn×n

sym , real numbers W 01 ,W

02 ∈ R, and a

positive definite fourth order tensor C : Mn×nsym → Mn×n

sym , let

Wj(E) =1

2(E − Ej) : C(E − Ej) +W 0

j

for E ∈ Mn×nsym and j = 1, 2. Then, if E1 = E2+(a⊗b+b⊗a)/2 for a, b ∈

Rn the quasiconvex hull of W : Mn×nsym → R, E 7→ minW1(E),W2(E),

is convex and given by [K]

ϕ(E) =

W1(E) for W2(E) + γ ≤ W1(E),12(W2(E) +W1(E))− 1

4γ(W2(E)−W1(E))2 − 4

γ

for |W1(E)−W2(E)| ≤ γ,W2(E) for W1(E) + γ ≤ W2(E),

for E ∈ Mn×nsym and γ = 1

2(E1 −E2) : C(E1 −E2). There holds (H1) for

S(A) = Dϕ((A+AT )/2), A ∈ Mn×n, with r = 2, s = 0, and a constant0 < α that depends on C [CP2].

More physical examples in the context of non-convex minimizationare included in [L1, L2, R].


Example 3.5 (Linear Right-Hand Side). Given functions f ∈ Lq(Ω; Rm)and g ∈ Lq(ΓN ; Rm) a typical linear right-hand side reads, for u, v ∈W 1,p(Ω; Rm),

J(u; v) =

∫Ω

f · v dx+

∫ΓN

g · v ds

where ΓN is a (possibly empty) part of ∂Ω. Note that J is independentof u and hence does not satisfy (H4).

Example 3.6 (linear Low-Order Terms). The derivative J = DΨ of astrictly convex low-order term Ψ in a model situation of [CP1] reads,for u, v ∈ W 1,p(Ω; Rm),

J(u; v) =

∫Ω

u · v ds

and satisfies (H4) for m = M = 1.

4. Proof of Theorem 2.1

The proof of Theorem 2.1 extends a technique from [CP1]. Fromthere we quote the first lemma.

Lemma 4.1. Suppose (H1)-(H2) and |Ω|s/p + |u|sW 1,p(Ω) + |uh|sW 1,p(Ω) ≤c1α. Then

‖δ‖rLq(Ω) ≤ c1

∫Ω

δ : Dedx.

Proof. The proof follows (in different notation) the arguments that leadto formula (3.7) in [CP1] and is hence omitted.

Direct algebra and (H3) imply the following result.

Lemma 4.2. Suppose (H2)-(H3) and eh ∈ AD,h. Then

2

∫Ω

δ : Dedx+β(‖e‖Xh

)≤ 2

∫Ω

δ : D(e− eh) dx+β∗(2B‖e− eh‖Yh

)+ 2

(Jh(u; eh)− J(u; eh)

).

Proof. The two identities in (H2) with v = eh = vh ∈ AD,h ⊆ AD yield∫Ω

δ : Dedx+ J(u; e)− Jh(uh; e)

=

∫Ω

δ : D(e− eh) dx+ J(u; e− eh)− Jh(uh; e− eh).


The differences on the left- and right-hand side are estimated with thefirst and second inequality of (H3) after inserting Jh(u; e) and Jh(u; e−eh), respectively, where v = e− eh. Hence,∫

Ω

δ : Dedx+ β(‖e‖Xh

)+

(J(u; e)− Jh(u; e)

)≤

∫Ω

δ : D(e− eh) dx

+B‖e‖Xh‖e− eh‖Yh

+(J(u; e− eh)− Jh(u; e− eh)

).

The definition of β∗ shows st ≤ β(s) + β∗(t) which, for s = ‖e‖Xhand

t = 2B‖e− eh‖Yh, results in

2B‖e‖Xh‖e− eh‖Yh

≤ β(‖e‖Xh

)+ β∗

(2B‖e− eh‖Yh

).

The combination of the last two estimates proves the lemma.

Lemma 4.3. Suppose (H1)-(H2) and let c2 := 2r′cr′−1

1 /r′. Then

2

∫Ω

δ : D(e− eh) dx ≤ (1/r)

∫Ω

δ : Dedx+ c2|e− eh|r′

W 1,p(Ω).

Proof. Holder’s and Young’s inequality show

2

∫Ω

δ : D(e− eh) dx ≤ ‖δ‖rLq(Ω)/(rc1) + 2r′c

r′/r1 |e− eh|r

′

W 1,p(Ω)/r′.

The assertion then follows from Lemma 4.1.

Proof of Theorem 2.1. This follows from Lemma 4.1, 4.2, and 4.3.

5. Finite Element Discretization

Let T be a regular triangulation of Ω into triangles (n = 2) ortetrahedra (n = 3) in the sense of [BS], i.e. no hanging nodes, thedomain is matched exactly, Ω = ∪T∈T T , and T satisfies the maximumangle condition. The extremal points of T ∈ T are called nodes and Ndenotes the set of all such nodes; K := N\∂Ω is the subset of free nodes.The set of edges (n = 2) or faces (n = 3) E = convz1, ..., zn ⊆ ∂Tfor pairwise distinct z1, ..., zn ∈ N and T ∈ T is denoted as E . By EΩ

we denote the set of interior edges or faces, EΩ = E ∈ E : ∃T1, T2 ∈T , E = T1 ∩ T2. We assume that ∂Ω is matched exactly by edgeson ∂Ω which implies ∂Ω = ∪E∈ED

E for the set of boundary edgesED := E ∈ E : E ⊆ ∂Ω. Let Pk(ω) denote the set of algebraicpolynomials of (total) degree ≤ k regarded as scalar functions on ω.The set

Pk(T ) := vh ∈ L∞(Ω) : ∀T ∈ T , vh|T ∈ Pk(T )


consists of all (possibly discontinuous) T –elementwise polynomials ofdegree at most k. We define

S1(T ) :=P1(T ) ∩ C(Ω) and AD,h = S10 (T )m := S1(T )m ∩W 1,2

0 (Ω; Rm).

Supposing that uD is continuous on ∂Ω we choose uD,h ∈ S1(T )m withuD,h(z) = uD(z) for all z ∈ N ∩ ∂Ω and set

Ah := uD,h + S10 (T )m.

Let (ϕz : z ∈ N ) be the nodal basis of S1(T ), i.e. ϕz ∈ S1(T ) satisfiesϕz(x) = 0 if x ∈ N \ z and ϕz(z) = 1. We set hT := diam (T ) forall T ∈ T and hE := diam (E) for all E ∈ E and define a functionhT ∈ L0(T ) by hT |T := hT for T ∈ T . Abbreviate h := ‖hT ‖L∞(Ω).We will frequently assume that T is quasiuniform which implies thath ≈ ‖h−1

T ‖−1L∞(Ω).

We write Hs(U ; Rm) for W s,2(U ; Rm) for an open set U ⊆ Rn and

Hs(T ; Rm) =v ∈ L2(Ω; Rm) : ∀T ∈ T , v|T ∈ Hs

(int(T ); Rm

).

The elementwise application of the differential operators D2 (the ma-trix of all second order derivatives) and ∆ (the Laplace operator) to afunction v ∈ H2(T ; Rm) is denoted by D2

T v and ∆T v, respectively.For each edge E ∈ EΩ we choose a vector νE ∈ Rn (with selected and

then fixed orientation) with |νE| = 1 orthogonal to E.Assume v ∈ H1(Ω; Rm) ∩H2(T ; Rm), let E ∈ EΩ be such that E =

T+ ∩ T− for T+, T− ∈ T and suppose νE points from T+ to T−. Then,define [Dv] ∈ L2(E; Mm×n) by

[Dv] := (Dv|T+)|E − (Dv|T−)|E.

For a function φ ∈ C(∂Ω; Rm) such that φ|E ∈ H2(E; Rm) for allE ∈ ED, ∂2

Eφ/∂s2 is the edgewise second derivative of φ along ∂Ω;

H2(ED; Rm) denotes the set of all such functions φ.Throughout this paper we abbreviate inequalities A ≤ C B with

an h-independent constant C > 0 by A . B and A ≈ B replacesA . B . A. The constant C may well depend on the shape of theelements; e.g. hE ≈ hT for E ∈ E and T ∈ T with E ⊆ ∂T . Forinstance, the well-established trace inequality reads

(5.1) ‖φ‖2L2(∂T ) . h−1

T ‖φ‖2L2(T ) + hT‖Dφ‖2

L2(T )

for any T ∈ T and φ ∈ H1(T ; Rm).


6. Stabilisation via jumps of gradients

This section is devoted to the discrete problem (Ph) with Jh := J+ah

for the bilinear form

(6.1) ah : Xh × Yh → R, (v, w) 7→∑E∈EΩ

hγE

∫E

[Dv] : [Dw] ds.

Therein, the spaces Xh and Yh are arbitrary with

(6.2) Xh = Yh ⊆ W 1,p(Ω; Rm) ∩H3/2+ε(T ; Rm)

for some ε with 0 < ε ≤ 1/2. Then the traces of Dv and Dw on ∪EΩ

for v, w ∈ Xh = Yh belong to L2(∪EΩ). Notice that S1(T )m ⊆ Xh bute ∈ Xh is some additional (and strong) hypothesis on u and that wewill even suppose u ∈ H2(Ω; Rm).

Theorem 6.1. Suppose (H1), (H2), and (H4) and uD ∈ H2(ED; Rm).Moreover, assume that u ∈ H2(Ω; Rm) ∩W 1,p(Ω; Rm) and T is quasi-uniform. Then, there holds

limh→0

‖Du−Duh‖L2(Ω) = 0 for − 1 < γ < 3,

‖u− uh‖W 1,2(Ω) ≤ c3h1/2 for γ = 1.

The constant c3 > 0 depends on c1, c2, and upper bounds for ‖u‖H2(Ω),|uh|W 1,p(Ω), |u|W 1,p(Ω), and ‖∂2

EuD/∂s2‖L2(∂Ω).

Remark 6.1. Provided u ∈ (H2(T ; Rm) ∩ W 1,p(Ω; Rm)) \ H2(Ω; Rm)there holds ah(u, ·) 6≡ 0. Then, for γ = 5/2, the proof of Theorem 6.1below can be modified to obtain the estimate

‖u− uh‖W 1,2(Ω) . h1/8.

The proof of the theorem follows from the abstract estimate of Theo-rem 2.1 and the following lemmas. Throughout this section, abbreviate

|v|h := ‖hγ/2E [Dv]‖L2(∪EΩ) and ‖v‖2

Xh= ‖v‖2

Yh:= ‖v‖2

L2(Ω) + |v|2hfor v ∈ H3/2+ε(T ; Rm).

Lemma 6.1. If eh is the nodal interpolant of e ∈ C(Ω; Rm) then

‖e− eh‖Yh. ‖h(1+γ)/2

T D2T e‖L2(Ω).

Proof. This is an immediate consequence of the trace inequality (5.1)and standard error estimates of nodal interpolation.

Proposition 2.2 and Lemma 6.1 allow for the application of Theo-rem 2.1. The strong convergence, however, is obtained by a combina-tion with the following argument.


Lemma 6.2. There holds

|e|2W 1,2(Ω) . ‖e‖L2(Ω)‖∆T e‖L2(Ω)

+ |e|h(‖h(1−γ)/2

T De‖L2(Ω) + ‖h−(1+γ)/2T e‖L2(Ω)

)+ ‖h2

T ∂2EuD/∂s

2‖L2(∂Ω)

(‖u‖H2(Ω) + ‖h−1/2

T Duh‖L2(Ω)

).

Proof. We perform an integration by parts on each T ∈ T , use the esti-

mates ‖Du·ν‖L2(∂Ω) . ‖u‖H2(Ω) and ‖Duh·ν‖L2(∂Ω) . ‖h−1/2T Duh‖L2(Ω),

and employ Cauchy inequalities to verify

‖De‖2L2(Ω) =

∑T∈T

∫∂T

(De · ν) · e ds−∑T∈T

∫T

(∆e) · e dx

=∑E∈EΩ

∫E

([De] · νE

)· e ds−

∫Ω

(∆T e) · e dx+

∫∂Ω

(De · ν) · e ds

.( ∑

E∈EΩ

hγE‖[De]‖

2L2(E)

)1/2( ∑E∈EΩ

h−γE ‖e‖2

L2(E)

)1/2

+ ‖∆T e‖L2(Ω)‖e‖L2(Ω) +(‖u‖H2(Ω) + ‖h−1/2

T Duh‖L2(Ω)

)‖e‖L2(∂Ω)

= |e|h( ∑

E∈EΩ

h−γE ‖e‖2

L2(E)

)1/2+ ‖e‖L2(Ω)‖∆T e‖L2(Ω)

+(‖u‖H2(Ω) + ‖h−1/2

T Duh‖L2(Ω)

)‖e‖L2(∂Ω).

The trace inequality (5.1) yields∑E∈EΩ

h−γE ‖e‖2

L2(E) . ‖h−(1+γ)/2T e‖2

L2(Ω) + ‖h(1−γ)/2T De‖2

L2(Ω).

Nodal interpolation estimates on each E ∈ ED show

‖e‖L2(∂Ω) . ‖h2T ∂

2EuD/∂s

2‖L2(∂Ω).

The combination of the last three estimates concludes the proof.

Proof of Theorem 6.1. Notice that [Du]|E = 0 for all E ∈ EΩ so thatah(u, eh) = 0. It follows from Theorem 2.1 and Lemma 6.1 that

‖e‖2L2(Ω) + |e|2h . |e− eh|r/(r−1)

W 1,p(Ω) + β∗(2B‖e− eh‖Yh

)+ 2ah(u, eh)

. ‖hTD2T e‖

r/(r−1)Lp(Ω) + ‖h(γ+1)/2

T D2T e‖2

L2(Ω)

. hr/(r−1) + hγ+1 =: RHS2.


The combination of this with Lemma 6.2 and ‖∆T e‖L2(Ω), ‖u‖H2(Ω),‖∂2

EuD/∂s2‖L2(∂Ω), ‖Duh‖L2(Ω) . 1 yields

|e|2W 1,2(Ω) . RHS + ‖h2T ‖L∞(Ω)‖h−1/2

T ‖L∞(Ω)

+ RHS(‖h−(1+γ)/2

T ‖L∞(Ω)RHS + ‖h(1−γ)/2T ‖L∞(Ω)‖De‖L2(Ω)

).

Young’s inequality allows us to absorb ‖De‖L2(Ω) = |e|W 1,2(Ω) on theright-hand side and hence shows

|e|2W 1,2(Ω) . RHS + RHS2‖h−(1+γ)/2T ‖L∞(Ω)

+ RHS2‖h(1−γ)/2T ‖2

L∞(Ω) + ‖h2T ‖L∞(Ω)‖h−1/2

T ‖L∞(Ω).

Since r ≤ 2, hr/(r−1) . h2. With ‖h−1T ‖L∞(Ω) ≈ ‖hT ‖−1

L∞(Ω) we deduce

|e|2W 1,2(Ω) . h+ h(γ+1)/2 + h(3−γ)/2 + h(γ+1)/2 + h3−γ + h2 + h3/2.

This and ‖e‖L2(Ω) . h2 + hγ+1 prove Theorem 6.1.

Remark 6.2. If boundary conditions are imposed only on some part ΓD

of ∂Ω (and not on the entire boundary ∂Ω) one obtains an additionalterm ∫

∂Ω\ΓD

(De · ν) · e ds

which we failed to control.

7. Stabilisation via distances to averages of gradients

This section is devoted to a stabilisation Jh = J + ah with distancesto averages of gradients, i.e.

(7.1) ah(v, w) :=

∫Ω

hγ−1T (Dv − ADv) : (Dw − ADw) dx

for γ ∈ R, v, w ∈ W 1,p(Ω; Rm), and for the averaging operator

A : L2(Ω; Mm×n) → S1(T )m×n, p 7→ Ap :=∑z∈N

|ωz|−1

∫ωz

p dxϕz.

Here, for each node z ∈ N , ωz = x ∈ Ω : ϕz(x) > 0 denotes its patchof area or volume |ωz|. Let Xh = Yh be as in Section 6. For v ∈ Xh weabbreviate

‖|v|‖2h = ah(v, v) = ‖h(γ−1)/2

T (Dv − ADv)‖2L2(Ω)

and define ‖ · ‖Xh= ‖ · ‖Yh

by

‖v‖2Xh

= ‖v‖2Yh

= ‖v‖2L2(Ω) + ‖|v|‖2

h.


Theorem 7.1. Under the hypotheses of Theorem 6.1 there holds

limh→0

‖Du−Duh‖L2(Ω) = 0 for − 1 < γ < 3,

‖u− uh‖W 1,2(Ω) ≤ c4h1/2 for γ = 1.

Remark 7.1. Provided u ∈ (H2(T ; Rm)∩W 1,p(Ω; Rm))\H2(Ω; Rm) andγ = 5/2, one can prove

‖u− uh‖W 1,2(Ω) . h1/8.

The following lemma shows that the stabilisation defined by (7.1) isequivalent to the one discussed in the previous section and will be usedto reduce the proof of Theorem 7.1 to the one of Theorem 6.1. Thesemi-norm | · |h is defined as in the previous section.

Lemma 7.1 ([C]). For vh ∈ S1(T )m there holds |vh|h ≈ ‖|vh|‖h.

Proof of Theorem 7.1. Let eh denote the nodal interpolant of e ∈C(Ω; Rm). Theorem 2.1 and Proposition 2.2 show

‖e‖2L2(Ω) + ‖|e|‖2

h . |e− eh|r/(r−1)

W 1,p(Ω)

+ ‖e− eh‖2L2(Ω) + ‖|e− eh|‖2

h + 2ah(u, eh).

Lemma 7.1 shows

|e|h ≤ |eh|h + |e− eh|h . ‖|eh|‖h + |e− eh|h. ‖|e|‖h + ‖|e− eh|‖h + |e− eh|h.

Nodal interpolation estimates and continuity of A then imply

|e|2h . ‖|eh|‖2h + hγ+1.

We employ Holder’s inequality, Young’s inequality, and nodal interpo-lation estimates to verify for % > 0

ah(u, eh) . ‖|u|‖2h + %‖|eh|‖2

h . ‖|u|‖2h + %‖|e|‖2

h + %‖|e− eh|‖2h

. ‖|u|‖2h + %‖|e|‖2

h + hγ+1.

Using∑

z∈N ϕz = 1, we deduce

‖|u|‖2h =

∑z∈N

∫Ω

hγ−1T ϕz(Du− pz)(Du− ADu) dx

≤(∑

z∈N

‖h(γ−1)/2T ϕ1/2

z (Du− pz)‖2L2(Ω)

)1/2

‖h(γ−1)/2T (Du− ADu)‖L2(Ω),


where pz = |ωz|−1∫

ωzDudx for all z ∈ N . Poincare’s inequality and

|ϕz| ≤ 1 show∑z∈N

‖h(γ−1)/2T ϕ1/2

z (Du− pz)‖2L2(Ω) . ‖h(γ+1)/2

T D2u‖2L2(Ω).

The combination of the preceding three estimates proves

‖e‖2L2(Ω) + |e|2h . h2 + hγ+1.

The assertions of the theorem then follow with Lemma 6.2 and thearguments of the proof of Theorem 6.1.

8. Stabilisation via gradients

This section is devoted to a stabilisation Jh = J +ah with gradients,i.e. with some γ > 0 and

(8.1) ah(v, w) = hγ

∫Ω

Dv : Dw dx

for all v, w ∈ Xh = Yh = W 1,p(Ω; Rm). For v ∈ Xh we define

‖v‖2Xh

= ‖v‖2Yh

:= ‖v‖2L2(Ω) + hγ‖Dv‖2

L2(Ω).

Theorem 8.1. Suppose (H1), (H2), and (H4). Assume that T isquasiuniform and u ∈ W 1,p(Ω; Rm)∩H1+s(Ω; Rm) for some s ∈ (1/2, 1].Then, there holds

limh→0

‖Du−Duh‖L2(Ω) = 0 for γ ∈ (2(1− s), 2s),

‖u− uh‖W 1,2(Ω) ≤ c5hs−1/2 for γ = 1.

The constant c5 > 0 depends on c1, c2, and upper bounds for ‖u‖H1+s(Ω),|uh|W 1,p(Ω), |u|W 1,p(Ω).

Proof. Proposition 2.2 and Theorem 2.1 prove

‖e‖2L2(Ω)+h

γ‖De‖2L2(Ω) . |e− eh|r/(r−1)

W 1,p(Ω) + ‖e− eh‖2L2(Ω)

+ hγ‖D(e− eh)‖2L2(Ω) + ah(u, eh)

(8.2)

for the nodal interpolant eh ∈ S10 (T )m of e ∈ C(Ω; Rm). Standard

estimates on nodal interpolation in H1+s(Ω) and r/(r − 1) ≥ 2 imply

|e− eh|r/(r−1)

W 1,p(Ω) + ‖e− eh‖2L2(Ω) + hγ‖D(e− eh)‖2

L2(Ω)

. h2s + h2+2s + hγ+2s.


If u ∈ H2(Ω; Rm) then integration by parts and eh = 0 on ∂Ω show

hγ

∫Ω

Du : Deh dx ≤ hγ‖u‖H2(Ω)‖eh‖L2(Ω).

Holder’s inequality and an elementwise inverse estimate verify

hγ

∫Ω

Du : Deh dx . hγ−1‖u‖H1(Ω)‖eh‖L2(Ω).

Interpolation of the last two estimates yields

ah(u, eh) = hγ

∫Ω

Du : Deh dx . hγ−(1−s)‖u‖H1+s(Ω)‖eh‖L2(Ω).

We further estimate

ah(u, eh) . hγ−(1−s)‖u‖H1+s(Ω)‖eh‖L2(Ω)

≤ hγ−(1−s)‖u‖H1+s(Ω)‖e− eh‖L2(Ω) + hγ−(1−s)‖u‖H1+s(Ω)‖e‖L2(Ω).

Nodal interpolation estimates and Young’s inequality imply for % > 0

ah(u, eh) . hγ−(1−s)+1+s + h2γ−2(1−s) + %‖e‖2L2(Ω).

The combination with (8.2) shows, after absorbing ‖e‖L2(Ω) on theright-hand side,

‖e‖2L2(Ω) + hγ‖De‖2

L2(Ω) . h2s + h2γ−2(1−s).

The following theorem states that the stabilisation scheme (8.1) isin fact the scheme of [NW] in 1D (up to a lumped integration of theright hand side f).

Theorem 8.2. Let n = m = 1, Ω := (0, 1), A = AD := W 1,p0 (0, 1),

J(u; v) :=

∫ 1

0

uv dx and Jh(uh; vh) :=1

2

∑z∈K

hzuh(z)vh(z)

for u, v ∈ W 1,p0 (0, 1) and uh, vh ∈ Ah = AD,h := S1

0 (T ). Then, for alluh, vh ∈ Ah, there holds

Jh(uh; vh) = J(uh; vh) +1

6

∫ 1

0

h2TDuhDvh dx.

Proof. Let 0 = z0 < z1 < ... < zm+1 = 1 be such that N =z0, z1, ..., zm+1 and set hj := zj − zj−1 for j = 1, ...,m + 1 so that


hzj= hj +hj+1 for j = 1, ...,m. Elementary calculations with vh(z0) =

vh(zm+1) = 0 show

J(uh; vh) =1

6

m+1∑j=1

hj

(2uh(zj−1)vh(zj−1) + uh(zj−1)vh(zj)

+ uh(zj)vh(zj−1) + 2uh(zj)vh(zj)),

Jh(uh; vh) =1

2

m+1∑j=1

hj

(uh(zj−1)vh(zj−1) + uh(zj)vh(zj)

).

Hence, there holds

Jh(uh; vh)− J(uh; vh)

=1

6

m+1∑j=1

hj

(uh(zj−1)vh(zj−1)− uh(zj−1)vh(zj)

− uh(zj)vh(zj−1) + uh(zj)vh(zj))

=1

6

m+1∑j=1

hj

(uh(zj)− uh(zj−1)

)(vh(zj)− vh(zj−1)

)=

1

6

∫ 1

0

h2TDuhDvh dx.

The parameter γ = 2 is critical in Theorem 8.1 and excluded inour analysis. In fact, the arguments in [NW] are quite different andrestricted to a model scenario in 1D.

9. Strong convergence in the scalar 2-well problem

In case of the 2-well energy from Example 3.3 and n ≥ 2, m = 1we can weaken (H4), i.e. the uniform monotonicity of J can in fact bereplaced by monotonicity.

(H5). Suppose that there exists B ≥ 0 such that, for v ∈ W 1,p(Ω),

0 ≤ J(u; e)− J(uh; e),

J(u; v)− J(uh; v) ≤ B‖e‖L2(Ω)‖v‖L2(Ω).

We suppose that Jh := J + ah with ah as in (6.1), (7.1), or (8.1).

Theorem 9.1. Suppose n ≥ 2 and m = 1. Let S = Dϕ with ϕ as inExample 3.3. Suppose (H5) and uD ∈ H2(ED; R). Assume that T isquasiuniform and u ∈ H2(Ω) ∩W 1,p(Ω). Then, there holds

‖u− uh‖W 1,2(Ω) ≤ c6h1/2 for γ = 1.


The constant c6 > 0 depends on c1, c2, and upper bounds for ‖u‖H2(Ω),|uh|W 1,p(Ω), |u|W 1,p(Ω), and ‖∂2

EuD/∂s2‖L2(∂Ω).

The proof of the theorem follows from the following lemma and theestimates of the previous sections.

Lemma 9.1. Let n ≥ 2 and let ϕ be as in Example 3.3 and S = Dϕ.Suppose eh ∈ H1(Ω) satisfies eh = 0 on ∂Ω. Then, there holds

‖e‖2L2(Ω) .

∫Ω

δ : Dedx+ ‖e− eh‖2L2(Ω) + ‖D(e− eh)‖2

L2(Ω).

Proof. Proposition 3 in [CP1] ensures the existence of some a ∈ Rn

with |a| = 1 such that

‖a ·De‖2L2(Ω) .

∫Ω

δ ·Dedx.

A fine version of Friedrichs’ inequality (which follows from the onedimensional Friedrichs inequality) proves

‖eh‖L2(Ω) . ‖a ·Deh‖L2(Ω).

Two applications of the triangle inequality and the last two estimatesprove the lemma.

Proof of Theorem 9.1. Proposition 2 and Theorem 2 in [CP1] prove(H1)-(H2). Setting ‖v‖2

Xh= ‖v‖2

Yh:= ah(v, v) we observe that the first

estimate in (H3) is satisfied. Instead of the second estimate in (H3) wehave

Jh(u; v)− Jh(uh; v) ≤ ‖e‖L2(Ω)‖v‖L2(Ω) + ‖e‖Xh‖v‖Yh

for all v ∈ Yh. This and Lemma 9.1 imply the estimate of Theorem 2.1.Hence,

‖e‖2L2(Ω) + ah(e, e) . |e− eh|r/(r−1)

W 1,p(Ω) + ‖e− eh‖2L2(Ω)

+ ‖D(e− eh)‖2L2(Ω) + ah(e− eh, e− eh) + 2ah(u, eh)

for eh ∈ S10 (T ). The estimate of the theorem then follows as in the

proofs of Theorem 6.1, 7.1, and 8.1 for ah defined by (6.1), (7.1), and(8.1), respectively.

Acknowledgements. The support by the DFG through the priority program1095 “Analysis, Modelling and Simulation of Multiscale Problems” is thank-fully acknowledged. This work was initiated when all authors were membersor guests at the Graduiertenkolleg 357 “Efficient Algorithms and Multiscale


Problems” in Kiel, Germany, continued at the Vienna University of Tech-nology, Austria, and finished while all the authors enjoyed the hospitalityof the Newton Institute, Cambridge, England, UK, during the programmeComputational Challenges in Partial Differential Equations.

References

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[Y] Young, L.C. (1937), Generalized curves and the existence of an attained ab-solute minimum in the calculus of variations. Comptes Rendues de la Societedes Sciences et des Lettres de Varsovie, classe III 30, 212-234.

Mathematisches Seminar, Christian-Albrechts-Universitat zu Kiel,Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany.

E-mail address: [email protected]

Humboldt-Universitat zu Berlin, Unter den Linden 6, 10099 Berlin.E-mail address: [email protected]

University of Warwick, Mathematics Institute, Coventry, CV4 7A,England.

E-mail address: [email protected]

Department of Mathematics, ETHZ, CH-8092 Zurich, Switzerland.E-mail address: [email protected]

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