A posteriori error analysis of stabilised FEM for degenerate convex minimisation problems under weak regularity assumptions

Background

The discretisation of degenerate convex minimisation problems experiences numerical difficulties with a singular or nearly singular Hessian matrix.

Methods

Some discrete analog of the surface energy in microstrucures is added to the energy functional to define a stabilisation technique.

Results

This paper proves (a) strong convergence of the stress even without any smoothness assumption for a class of stabilised degenerate convex minimisation problems. Given the limitted a priori error control in those cases, the sharp a posteriori error control is of even higher relevance. This paper derives (b) guaranteed a posteriori error control via some equilibration technique which does not rely on the strict Galerkin orthogonality of the unperturbed problem. In the presence of L² control in the original minimisation problem, some realistic model scenario with piecewise smooth exact solution allows for strong convergence of the gradients plus refined a posteriori error estimates. This paper presents (c) an improved a posteriori error control in this interface problem and so narrows the efficiency reliability gap.

Conclusions

Numerical experiments illustrate the theoretical convergence rates for uniform and adaptive mesh-refinements and the improved a posteriori error control for four benchmark examples in the computational microstructures.

Infimising sequences of variational problems with non-quasiconvex energy densities, in general, develop finer and finer oscillations with no classical limit in Sobolev function spaces called microstructure [1–6]. Those oscillations cause difficulty to numerical methods because fine grids are necessary to resolve such oscillations which results in ineffective and tricky mesh-depending computations. Strong convergence of gradients of infimising sequences of the non-quasiconvex problem is impossible.

Relaxation techniques replace the nonconvex energy density by its (semi-)convex hull and lead to a macroscopic model. Since the convexified energy density obtained by this method, in general, lacks strict convexity, numerical algorithms might encounter situations where the Hessian matrix is singular. For instance, the Newton minimisation algorithm fails on the convexified three-well problem of Subsection ‘Three-well benchmark’ below. Applications of relaxation techniques include models in computational microstructure [5–7], some optimal design problems [8, 9], the nonlinear Laplacian [10] (where the Hessian can become arbitrarily ill-conditioned in spite of its strict convexity) and elastoplasticity [1].

Stabilisation techniques regularise the energy term by an additional positive semidefinite stabilisation function. The paper [11] discusses several choices of such stabilisation functions for P₁ conforming finite elements and quasiuniform meshes. It turns out that stabilisation can ensure strong convergence of the strain approximations under particular circumstances. A particular stabilisation in [12] leads to strong convergence even on unstructured grids but is still restricted to unrealistically smooth solutions. This paper studies the stabilisation technique of [12] and addresses the question of convergence (i) without extra regularity assumptions, (ii) in a realistic scenario called model interface problem, and (iii) establishes an a posteriori error control.

The stabilisation leads to improved condition numbers of the Hessian matrix and to reduced errors if the numerical solvers fail without stabilisation. Figure 1 shows the convergence of the discrete stress σ _ℓ of the three-well benchmark corresponding to the discrete minimisers of the energy E_{\ell }\left(v_{\ell }\right)=E\left(v_{\ell }\right)+C/2{{\parallel |v}_{\ell }\parallel |}_{\ell }^2. The errors are plotted for computations with uniform mesh refinements with various solver tolerances in the discrete minimisation procedure at a fixed triangulation and values of C, cf. Section ‘Numerical experiments’ for details on the MATLAB implementation. Without stabilisation, the convergence stagnates with a moderate tolerance of 10^-5 which becomes visible as a “plateau” in Figure 1. The Newton solver even aborts prematurely due to the singular Hessian. In conclusion, stabilisation enables higher accuracies in numerical examples.

Impact of the stabilisation on the error. Error of the stabilised stress σ _ℓ with coefficients C = 0, 10^-4,1 of the stabilisation and various tolerances tol = 10^-5, 10^-6, 10^-8 of the Newton solver.

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For \beta \ge 0 the convex energy functional assumes the form

Assume that W is convex with quadratic growth so that there exist minimisers u\in H_0^1\left(\mathrm{\Omega }\right); below p-th order growth is included while p = 2 throughout this simplifying introduction. Given a sequence of shape-regular triangulations {\left({\mathcal{T}}_{\ell }\right)}_{\ell \in {\mathbb{N}}_0}[13], let u _ℓ minimise the stabilised discrete energy

amongst all conforming P₁ finite element functions v _ℓ on {\mathcal{T}}_{\ell }, where [Dv _ℓ ] _F is the jump of the gradient Dv _ℓ along the interior side F, written F\in {\mathcal{F}}_{\ell }\left(\mathrm{\Omega }\right), and H _ℓ  := maxT h _T is the maximal diameter h _T of all simplices T\in {\mathcal{T}}_{\ell }.

Section ‘Global convergence’ verifies the strong convergence of the discrete solution u _ℓ and its stress σ _ℓ  := DW(Du _ℓ ) to their respective continuous conterparts,

Section ‘A posteriori error estimates’ presents a novel application of [14–17] to nonlinear problems. For the L² projection Π _ℓ onto the space of piecewise P₀ functions, any Raviart-Thomas function {\tau }_{\ell }\in {\mathit{\text{RT}}}_0\left({\mathcal{T}}_{\ell }\right) satisfies

This error bound holds for any discrete displacement u _ℓ that satisfies the boundary conditions; the point is that inexact solve is included — there is no Galerkin orthogonality required. The drawback is to minimise the expression on the right-hand side with respect to τ _ℓ in order to obtain a sharp error bound. This is a particular selection: degenerate convex minimisation problems do not allow for a control of ||u-u_{\ell }|{|}_{H^1\left(\mathrm{\Omega }\right)} and may even face multiple exact or discrete solutions while the discrete minimum of E _ℓ is unique. However, in some results of this paper, either W or the lower-order terms lead to some control over ||u-u_{\ell }|{|}_{H^2\left(\mathrm{\Omega }\right)} and the selection via stabilisation is correct.

Phase transition problems motivate the investigation of scenarios with a smooth solution u up to a one-dimensional interface \mathrm{\Gamma }\subset \overline{\mathrm{\Omega }}[18]. Section ‘Refined analysis for an interface model problem’ proves that such problems allow even for strong convergence of the gradients for any unique solution u in W^1,∞(Ω)∩H²(Ω∖Γ) [19]. This result also leads to an improvement of the a posteriori error control of the discrete stresses and narrows the efficiency-reliability gap; the efficiency-reliability gap is the difference of the convergence rates of the guaranteed upper a posteriori error bound and the guaranteed lower a posteriori error bound.

Section ‘Numerical experiments’ complements the theoretical findings with numerical experiments to provide empirical evidence of the improved error control. The stabilisation technique competes in four benchmark examples, with and without known exact solution, for uniform and two different mesh-refining algorithms for the explicit residual-based error estimator of [7] and with an averaging-type error estimator of ([18], (1.11)).

Standard notation on Lebesgue and Sobolev spaces is employed throughout this paper and a ≲ b abbreviates a ≤ C b with some generic constant 0 < C < ∞ independent of crucial parameters (like the mesh-size on level ℓ); a ≈ b means a ≲ b ≲ a. Furthermore, A:B abbreviates the matrix inner product that corresponds to the Frobenius norm.

Based on the convergence results for unstructured grids, this paper will develop reliable error estimators for a class of stabilised convex minimisation problems described in the sequel. Let \mathrm{\Omega }\subset {\mathbb{R}}^n be a bounded Lipshitz domain with polygonal boundary for n = 2 or 3. Given a continuous convex energy density W:{\mathbb{R}}^{m\times n}\to \mathbb{R}, g,f\in L^2(\mathrm{\Omega };{\mathbb{R}}^m), \beta \ge 0, and v\in W^{1,p}(\mathrm{\Omega };{\mathbb{R}}^m) with 2 ≤ p < ∞ and m = 1, …, n, the energy is given by (1.1).

Throughout this paper, the energy density W\in {\mathcal{C}}^1({\mathbb{R}}^{m\times n};\mathbb{R}) satisfies (2.1)–(2.2) for parameters 1 < r ≤ 2, 0 ≤ s < ∞ and s + r + p ≤ r p. The two-sided growth condition reads

The convexity control assumption reads, for all F_1,F_2\in {\mathbb{R}}^{m\times n},

The proof of Theorem 2 in [7] shows that (2.2) is crucial for the uniqueness of the stress tensor DW(Du).

Given Dirichlet data u_D\in W^{2,p}(\mathrm{\Omega };{\mathbb{R}}^m)\cap H^2(\mathrm{\partial \Omega };{\mathbb{R}}^m) for the set of admissible functions \mathcal{A}:=u_D+V:=u_D+W_0^{1,p}(\mathrm{\Omega };{\mathbb{R}}^m), the continuous (convex) model problem reads

A finite element approximation of (2.3) is based on a family of regular triangulations {\left({\mathcal{T}}_{\ell }\right)}_{\ell \in {\mathbb{N}}_0} of the domain Ω into simplices in the sense of Ciarlet [13] (e.g., for n = 2, two non-disjoint triangles of {\mathcal{T}}_{\ell } share either a common edge or a common node). The set of sides {\mathcal{F}}_{\ell } consists of edges (for n = 2) or faces (for n = 3) of {\mathcal{T}}_{\ell } and is split into the union of the sets of all interiour sides {\mathcal{F}}_{\ell }\left(\mathrm{\Omega }\right) and of all boundary sides {\mathcal{F}}_{\ell }\left(\mathrm{\partial \Omega }\right).

For latter reference, define the diameter h _T :=diamT of a triangle (or tetrahedron) T\in {\mathcal{T}}_{\ell } and the size h _F  := diamF of a side F\in {\mathcal{F}}_{\ell }. The mesh size functionh_{\ell }:\mathrm{\Omega }\to {\mathbb{R}}_{>0} is given by

The global mesh size will be abbreviated by H_{\ell }:={\parallel h}_{\ell }{\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}. We presume the family {\left({\mathcal{T}}_{\ell }\right)}_{\ell \in {\mathbb{N}}_0} to be shape-regular so that h _F  ≈ h _T for all T\in {\mathcal{T}}_{\ell }, F\in {\mathcal{F}}_{\ell } and F ⊂ T.

The space of {\mathcal{T}}_{\ell }-piecewise polynomials of degree \le k\in {\mathbb{N}}_0 is P_k\left({\mathcal{T}}_{\ell }\right). The nodal interpolation I_{\ell }w\in P_1\left({\mathcal{T}}_{\ell }\right)\cap C\left(\overline{\mathrm{\Omega }}\right) of w\in C\left(\overline{\mathrm{\Omega }}\right) is given by I _ℓ w(z) = w(z) for all nodes z. Let furthermore Π _ℓ w be the L² projection of w ∈ L²(Ω) onto P_0\left({\mathcal{T}}_{\ell }\right), and {\text{osc}}_{\ell ,q}\left(w\right):={\parallel h}_{\ell }(\text{id}-{\mathrm{\Pi }}_{\ell })w{\parallel }_{L^q\left(\mathrm{\Omega }\right)} be the oscillation of w ∈ L^q(Ω) for 2 ≤ q ≤ ∞ with respect to the triangulation {\mathcal{T}}_{\ell }. The symbol id denotes the identity operator. Let u_D,ℓ = I _ℓ u _D , and

Given a function v on Ω which is possibly discontinuous along some side F\in {\mathcal{F}}_{\ell }\left(\mathrm{\Omega }\right) shared by the two elements T_± such that there exist traces from either sides, the jump of v along F reads

The stabilisation of [12] will be used throughout this paper with -1 < γ < ∞ and

The stabilised discrete problem reads

Convergence of gradients with a guaranteed convergence rate is shown in [12] under unrealistically high regularity assumptions. A comprehensive collection of the results in [12] is summarised in the following theorem.

Theorem 2.1.

([12]) Letu\in \mathcal{A}\cap H^{3/2+\epsilon }(\mathrm{\Omega };{\mathbb{R}}^m)be some solution of (2.3) for some ε > 0; let p^′and r^′be the Hölder conjugate of p and r, -1 < γ < 3, and set

Then the discrete solution u_{\ell }\in {\mathcal{A}}_{\ell } of (2.5) and the continuous and discrete stress \sigma :=\mathrm{D}W\left(\mathrm{D}u\right)\in L^{p^{\prime }}(\mathrm{\Omega };{\mathbb{R}}^{m\times n}) and {\sigma }_{\ell }:=\mathrm{D}W\left(\mathrm{D}{u}_{\ell }\right)\in P_0({\mathcal{T}}_{\ell };{\mathbb{R}}^{m\times n}) satisfy

Proof

This combines Lemma 3.5 and 4.1 – 4.2 plus Theorem 3.8 and 4.4 in [12].□

This section is devoted to the proof of a general convergence result without higher regularity assumptions. Let u\in \mathcal{A} and u_{\ell }\in {\mathcal{A}}_{\ell } solve the minimisation problem (2.3) and (2.5) and set σ := DW(Du) and σ _ℓ  := DW(Du _ℓ ). For the unstabilised approximation, the a priori error estimates of [7] plus a density argument prove convergence of

Note that β = 0 is permitted. Then, however, uniqueness of u and convergence of \parallel u-u_{\ell }{\parallel }_{L^2\left(\mathrm{\Omega }\right)}^2 are guaranteed. The point in the following result is that the stabilised approximation converges as well as |||u _ℓ ||| _ℓ  → 0 even for non-smooth or non-unique minimisers. Under special circumstances, uniqueness of u and the convergence \parallel u-u_{\ell }{\parallel }_{L^2\left(\mathrm{\Omega }\right)}\to 0 can be shown even for β = 0, e.g., in Example 3.3.

Theorem 3.1.

( Global Convergence) Provided{lim}_{\ell \to \infty }{H}_{\ell }=0 it holds

The proof is based on the following lemma.

Lemma 3.2.

The errors δ _ℓ  := σ - σ _ℓ and e _ℓ  := u - u _ℓ satisfy, for all v _ℓ  ∈vV _ℓ , that

Proof

The minimisation problems (2.3) and (2.5) are equivalent to their respective Euler-Lagrange equations, namely for v ∈ V and v _ℓ  ∈ V _ℓ ,

Algebraic transformations of the difference of these two equations lead to

It is shown in ([12], Lemma 3.5) that

Two Hölder inequalities on the right-hand side and absorbtions of {\parallel \delta }_{\ell }{\parallel }_{L^{p^{\prime }}\left(\mathrm{\Omega }\right)} and {\parallel e}_{\ell }{\parallel }_{L^2\left(\mathrm{\Omega }\right)} eventually conclude the proof. Further details are dropped for brevity.□

Proof of Theorem 3.1

Given any positive ε, the density of smooth functions in W_0^{1,p}(\mathrm{\Omega };{\mathbb{R}}^m) leads to some v_{\epsilon }\in \mathcal{D}(\mathrm{\Omega };{\mathbb{R}}^m) such that \parallel u-u_D-v_{\epsilon }{\parallel }_{W^{1,p}\left(\mathrm{\Omega }\right)}\lesssim \epsilon. Hence v _ℓ  := I _ℓ (v _ε  + u _D ) - u _ℓ  ∈ V _ℓ satisfies

Note that the nodal interpolation I _ℓ (v _ε  + u _D ) is well-defined since v _ε and u _D are assumed to be smooth. With ([12], Lemma 3.1 – 3.2) it follows that

Since \parallel ·{\parallel }_{L^2\left(\mathrm{\Omega }\right)}\lesssim \parallel ·{\parallel }_{W^{1,p}\left(\mathrm{\Omega }\right)}, this yields some {\ell }_0\in \mathbb{N} such that

A Cauchy inequality applied to the stabilisation norm proves

Substitute a _ℓ (u _ℓ , v _ℓ ) in Lemma 3.2 and add \frac{1}{2}\left|\right|\left|u_{\ell }\right||{|}_{\ell }^2 on both sides. This leads to

□

Example 3.3.

The two-well example from the computational benchmark [18] allows an estimate on {\parallel e}_{\ell }{\parallel }_{L^2\left(\mathrm{\Omega }\right)} even for β = 0. Let n = 2, let F_1:=-F_2:=(3,2)/\sqrt{13}, and let the energy density W be the convex hull of F ↦ |F - F₁|²|F - F₂|². That is

Then ([11], Lemma 9.1) proves, for all v _ℓ  ∈ V _ℓ , that

Therefore, the arguments of Lemma 3.2 lead to

This result can be used in the proof of Theorem 3.1 in order to obtain

Beyond the a posteriori error analysis of [7], the additional stabilisation term in the discretisation of this paper causes an additional difficulty in that the Galerkin orthogonality does not hold for the natural residual. Inspired from novell developments in the a posteriori error control of elliptic PDEs motivated by inexact solve [14–17], this section presents some guaranteed upper error bound for the discretisation at hand for any approximation u _ℓ which does not necessarily satisfy (3.2) exactly. Thereby inexact solve is included.

Let u\in \mathcal{A} solve (2.3) and let u_{\ell }\in {\mathcal{A}}_{\ell } be arbitrary. It is not assumed that u _ℓ solves the discrete problem (2.5); the following theorem holds regardless of this. Recall the definitions of osc_ℓ,q(·) and Π _ℓ from Section ‘Methods: Discretisation and Stabilisation’ and given σ := DW(Du) and σ _ℓ  := DW(Du _ℓ ), abbreviate

Theorem 4.1.

Given anyw_{\ell }\in W^{1,p}(\mathrm{\Omega };{\mathbb{R}}^m)with w _ℓ  = u - u _ℓ on the boundary ∂ Ω, and given any\tau \in H(\mathit{\text{div}},\mathrm{\Omega };{\mathbb{R}}^{m\times n}), it holds, for all 2 ≤ q ≤ p and for some constant ϰ known from ([12], Lemma 3.5), that

The constant ϰ depends on problem-specific data such as \parallel u{\parallel }_{W^{1,p}\left(\mathrm{\Omega }\right)} and the size of the domain Ω. Refer to the proof of Lemma 3.5 in [12] for details.

Before the proofs conclude this section, some practical choice of τ in Theorem 4.1 is discussed as some Raviart-Thomas finite element functions in

We suggest the computation (or an accurate approximation) of

and emphasise that any upper bound is allowed in Theorem 4.1. This leads to

The algorithm of ([20], Prop. 4.1) computes some w _ℓ from (id-I _ℓ )u _D with

(The proof of the second assertion is analogous to that of ([20], Prop. 4.1) and the first is an immediate consequence of the design of w _ℓ ). This and {\parallel e}_{\ell }-w_{\ell }{\parallel }_{W^{1,q}\left(\mathrm{\Omega }\right)}\lesssim 1 for bounded u _ℓ (i.e. solely {\parallel u}_{\ell }{\parallel }_{W^{1,p}\left(\mathrm{\Omega }\right)}\lesssim 1 is assumed) lead to the practical estimate μ _ℓ as a computable guaranteed upper bound of the left-hand side of Theorem 4.1. Since the minimisation of (4.1) is computationally intensive for q ≠ 2, Section ‘Numerical experiments’ actually computes an approximation of μ _ℓ , based on q = 2.

The choice τ = σ in Theorem 4.1 shows that the right-hand side is in fact optimal up to oscillations. The reliability-efficiency gap of [18] is visible here in that we have no further estimate on {\parallel u}_{\ell }{\parallel }_{W^{1,p}\left(\mathrm{\Omega }\right)}[7, 18]. However, additional smoothness assumptions on u may lead to refined estimates on the term {\parallel e}_{\ell }-w_{\ell }{\parallel }_{W^{1,q}\left(\mathrm{\Omega }\right)} (cf. Section ‘Refined analysis for an interface model problem’). The following result indicates that μ _ℓ is sharp in the sense that it converges with the correct convergence rate. This theorem employs the Fortin interpolation operator I_F,ℓ defined for \tau \in H(\mathit{\text{div}},\mathrm{\Omega })\cap L^t(\mathrm{\Omega };{\mathbb{R}}^n) with t > 2 by I_{\mathrm{F},\ell }\tau \in {\mathit{\text{RT}}}_0\left({\mathcal{T}}_{\ell }\right) and

Here and in the following, n _F denotes a unit normal vector of the side F; the direction of n _F arbitrary, but fixed for a given side F. For the improved regularity of stress in the class of degenerate convex minimisation problems at hand, we refer to [3, 21].

Theorem 4.2.

( Efficiency) If the exact stress σ is sufficiently regular such that its Fortin interpolant{\tau }_{\ell }=I_{\mathrm{F},\ell }\sigma \in {\mathit{\text{RT}}}_0({\mathcal{T}}_{\ell };{\mathbb{R}}^{m\times n})is defined, it holds

It is expected that \parallel (\text{id}-I_{\mathrm{F},\ell })\sigma {\parallel }_{L^{q^{\prime }}\left(\mathrm{\Omega }\right)}\lesssim H_{\ell }. This is shown in ([22], Prop. 3.6) for q^′ = 2 and therefore also holds for q^′ ≤ 2. Hence the right-hand side of the assertion of Theorem 4.2 converges with the (expected) optimal convergence rates.

Proof of Theorem 4.1.

Let ϰ be the reciprocal of c₁ in ([12], Lemma 3.5), which is also the multiplicative constant hidden in (3.3). Recall Young’s inequality, which reads \mathit{\text{ab}}\le a^r/r+b^{r^{\prime }}/r^{\prime } for a, b > 0. This, (3.3) and the continuous Euler-Lagrange equation (3.1) show, for v = e _ℓ  - w _ℓ  ∈ V, that

Hence {\text{Res}}_{\ell }\left(v\right):=-{\int }_{\mathrm{\Omega }}({\sigma }_{\ell }:\mathrm{D}v-{\mathrm{\Lambda }}_{\ell }·v)\mathrm{d}x satisfies

Let C_{q^{\prime }} denote the Poincaré constant of convex domains with respect to the W^{1,q^{\prime }} norm. The fundamental theorem of calculus on some one-dimensional arc shows that C _∞  ≤ 1. The paper [23] proves C₁ = 1 / 2. Hence, operator-interpolation arguments [24, 25] prove C_{q^{\prime }}\le {(1/2)}^{1/q^{\prime }}\le 1. The Poincaré inequality shows, for any 2 ≤ q ≤ p, that

For any \tau \in H(\mathit{\text{div}},\mathrm{\Omega };{\mathbb{R}}^{m\times n}), the Hölder and Poincaré inequalities show

Proof of Theorem 4.2

The triangle inequality yields

Since f = 2β(u - g)-divσ, the commutative property divI_F,ℓ = Π _ℓ div (cf. ([22], p. 129)) yields

This section is devoted for a model scenario from phase transition problems [18] with some solution u that is smooth outside some one-dimensional interface Γ. Suppose some (possibly non-unique) minimiser u of the continuous problem (2.3) satisfies u\in W^{1,\infty }(\mathrm{\Omega };{\mathbb{R}}^m)\cap W^{2,p}(\mathrm{\Omega }\setminus \mathrm{\Gamma };{\mathbb{R}}^m) for some finite union Γ of (n - 1) dimensional Lipschitz surfaces in \overline{\mathrm{\Omega }}. Since Ω has a Lipschitz boundary, this implies Lipschitz continuity of u on Ω. We refer to [19] for sufficient conditions for u\in W^{1,\infty }(\mathrm{\Omega };{\mathbb{R}}^m) and conclude that the remaining assumption u\in W^{2,p}(\mathrm{\Omega }\setminus \mathrm{\Gamma };{\mathbb{R}}^m) is the essential hypothesis expected in many interface problems. Let u_{\ell }\in {\mathcal{A}}_{\ell } be the (unique) minimiser of the discrete stabilised problem (2.5). In the following, also Γ = ∅ is permitted to extend previous results [12] for highly regular minimisers.

The following theorem leads to a priori convergence rates for the interface model problem. Thereby it recovers the results of [12] for problems with piecewise smooth exact solution.

We will abbreviate the set of all triangles that are touched by Γ as {\mathcal{T}}_{\ell }\left(\mathrm{\Gamma }\right):=\{T\in {\mathcal{T}}_{\ell }:\text{dist}(T,\mathrm{\Gamma })=0\}, its cardinality as \left|{\mathcal{T}}_{\ell }\left(\mathrm{\Gamma }\right)\right|, its union as {\mathrm{\Omega }}_{\mathrm{\Gamma },\ell }:=\text{int}(\bigcup {\mathcal{T}}_{\ell }(\mathrm{\Gamma }\left)\right) with volume |Ω_Γ,ℓ| and its complement as {\mathrm{\Omega }}_{\mathrm{\Gamma },\ell }^{\mathrm{C}}:=\mathrm{\Omega }\setminus \overline{{\mathrm{\Omega }}_{\mathrm{\Gamma },\ell }}.

Theorem 5.1.

Provided β > 0, it holds

Remark 5.2.

In the case of uniform mesh refinements we may expect \left|{\mathcal{T}}_{\ell }\right(\mathrm{\Gamma }\left)\right|\approx H_{\ell }^{1-n} and |Ω_Γ,ℓ| ≈ H _ℓ and Theorem 5.1 simplifies to

Proof.

With w _ℓ  = (id - I _ℓ )e _ℓ  = (id - I _ℓ )u, a Young inequality, (3.3) and ([12], Theorem 3.8) yield

Theorem 4.4.4 in [25] shows {\parallel w}_{\ell }{\parallel }_{L^2\left(\mathrm{\Omega }\right)}\lesssim {\parallel w}_{\ell }{\parallel }_{L^{\infty }\left(\mathrm{\Omega }\right)}\lesssim H_{\ell }|u{|}_{W^{1,\infty }\left(\mathrm{\Omega }\right)} and

Let {\omega }_F=\bigcup _{\begin{array}{c}T\in T_{\ell }\\ F\subset T\end{array}}T be the patch of a side F\in {\mathcal{F}}_{\ell }, and set {\mathcal{F}}_{\ell }\left(\mathrm{\Gamma }\right)=\{F\in {\mathcal{F}}_{\ell }(\mathrm{\Omega }):{\omega }_F\cap \mathrm{\Gamma }\ne \varnothing \} and {\mathcal{F}}_{\ell }^{\mathrm{C}}\left(\mathrm{\Gamma }\right)={\mathcal{F}}_{\ell }\left(\mathrm{\Omega }\right)\setminus {\mathcal{F}}_{\ell }\left(\mathrm{\Gamma }\right). Note that [Du] _F = 0 for F\in {\mathcal{F}}_{\ell }^{\mathrm{C}}\left(\mathrm{\Gamma }\right). Then

The first sum can be estimated as in the proof of ([12], Lemma 3.2), the second sum with