Best approximation property in the $W^1_{\infty }$ norm for finite element methods on graded meshes

Demlow, Alan; Leykekhman, Dmitriy; Schatz, Alfred H.; Wahlbin, Lars B.

doi:10.1090/s0025-5718-2011-02546-9

Cited by 42 publications

(38 citation statements)

References 38 publications

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“…We will denote Having chosen the constant c I large enough, local W 1,∞ -error estimates from [14, Corollary 1] yieldwhere I hφ denotes the Lagrange interpolant ofφ. Notice, according to [14, Remark 2], this inequality is only valid for any J = 0, . .…”

mentioning

confidence: 99%

Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes

Apel¹,

Mateos²,

Pfefferer³

et al. 2018

Mathematical Control &Amp; Related Fields

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The paper deals with finite element approximations of elliptic Dirichlet boundary control problems posed on two-dimensional polygonal domains. Error estimates are derived for the approximation of the control and the state variables. Special features of unconstrained and control constrained problems as well as general quasiuniform meshes and superconvergence meshes are carefully elaborated. Compared to existing results, the convergence rates for the control variable are not only improved but also fully explain the observed orders of convergence in the literature. Moreover, for the first time, results in non-convex domains are provided. of the adjoint state has a pole there, we getū ∈ H s (Γ) for all s < 3 2 . This regularity is determined by the larges convex angle and by the kinks due to the constraints.Unfortunately, this is not the whole truth. In exceptional cases, e. g. when the data enjoy certain symmetry, the leading singularity of type r 2/3 may not appear in the adjoint state. Instead, the solution may have a r 4/3 -singularity whose normal derivative has a r 1/3 -singularity which is not flattened by the projection Proj [a,b] . The control is less regular,ū ∈ H s (Γ) for all s < 5 6 . See Example 3.6 in [1]. Hence, dealing with these exceptional cases is not fun but necessary. If in the unconstrained case a stress intensity factor vanishes , i.e., the leading singularity does not occur, then the convergence result is still true, one may only see a better convergence in numerical tests. See Figure 3, right hand side and Remark 4.8. However, in the constrained case, the situation is the opposite. The exceptional case leads to the worstcase estimate. To deal with the "worst-case" and the "usual-case" in an unified way, we introduce in (2.4) some numbers related to the singular exponents.We distinguish two cases for the investigation of the discretization errors. After proving a general result in Section 3 we study the unconstrained case in Section 4 and the constrained case in Section 5. We focus on quasi-uniform meshes and distinguish general meshes and certain superconvergence meshes. In order not to overload the present paper, we postpone the study of graded meshes to [2]. The numerical tests in Section 6 confirm the theoretical results.The study of error estimates for Dirichlet control problems posed on polygonal domains can be traced back to [9], where a control constrained problem governed by a semilinear elliptic equation posed in a convex polygonal domain is studied. An order of convergence of h s is proved for all s < min(1, π/(2ω 1 )), where ω 1 is the largest interior angle, in both the control and the state variable. Later, in [18], it is proven that for unconstrained linear problems posed on convex domains, the state variable exhibits a better convergence property. The corresponding proof is based on a duality argument and estimates for the controls in weaker norms than L 2 (Γ). However, to the best of our knowledge, the argumentation is restricted to unconstrained problems. For the error of t...

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mentioning

confidence: 99%

Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes

Apel¹,

Mateos²,

Pfefferer³

et al. 2018

Mathematical Control &Amp; Related Fields

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“…The second correction term is local, limited to one element and not to be passed down to the next element. The following two

W_{\infty}^{1}

estimates (cf., , , Theorem 5.5.2], , Theorem 3.1]) will be used in the rest of our proof:

‖ \nabla (p - p_{h}) ‖_{L^{\infty}} ≲ h^{k} ‖ p ‖_{W_{\infty}^{k + 1}}, ‖ \nabla p - G_{h} p_{h} ‖_{L^{\infty}} ≲ h^{k + 1} ‖ p ‖_{W_{\infty}^{k + 2}} .

The recovered super‐convergence is originally proved for on an interior domain. We assume it near the boundary.…”

Section: Element By Element Conserving‐flux Recoverymentioning

confidence: 99%

“…The following two W 1 ∞ estimates (cf., [48,49], [50,Theorem 5.5.2], [30, Theorem 3.1]) will be used in the rest of our proof:…”

Section: E Convergence Theorymentioning

confidence: 99%

A postprocessed flux conserving finite element solution

Zhang

Zou

2017

Numerical Methods Partial

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We propose a local postprocessing method to get a new finite element solution whose flux is conservative element‐wise. First, we use the so‐called polynomial preserving recovery (postprocessing) technique to obtain a higher order flux which is continuous across the element boundary. Then, we use special bubble functions, which have a nonzero flux only on one face‐edge or face‐triangle of each element, to correct the finite element solution element by element, guided by the above super‐convergent flux and the element mass. The new finite element solution preserves mass element‐wise and retains the quasioptimality in approximation. The method produces a conservative flux, of high‐order accuracy, satisfying the constitutive law. Numerical tests in 2D and 3D are presented.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1859–1883, 2017

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“…Later the result was extended to convex polyhedral domains with some restriction on angles in [2]. This restriction was removed in [12] and even extended to certain graded meshes in [6]. For parabolic problems similar results are rather scarce.…”

Section: Introductionmentioning

confidence: 99%

“…the Ritz projection R h : 6) and the usual nodal interpolation operator i h : C 0 (Ω) → V h with usual approximation properties (cf., e. g., [5, Theorem 3.…”

Section: Introductionmentioning

confidence: 99%

Global and Interior Pointwise best Approximation Results for the Gradient of Galerkin Solutions for Parabolic Problems

Leykekhman¹,

Vexler²

2017

SIAM J. Numer. Anal.

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Abstract. In this paper we establish best approximation property of fully discrete Galerkin solutions of second order parabolic problems on convex polygonal and polyhedral domains in the L ∞ (I; W 1,∞ (Ω)) norm. The discretization method consists of continuous Lagrange finite elements in space and discontinuous Galerkin methods of arbitrary order in time. The method of the proof differs from the established fully discrete error estimate techniques and uses only elliptic results and discrete maximal parabolic regularity for discontinuous Galerkin methods established by the authors in [15]. In addition, the proof does not require any relationship between spatial mesh sizes and time steps. We also establish interior best approximation property that shows more local dependence of the error at a point.

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Best approximation property in the $W^1_{\infty }$ norm for finite element methods on graded meshes

Cited by 42 publications

References 38 publications

Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes

Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes

A postprocessed flux conserving finite element solution

Global and Interior Pointwise best Approximation Results for the Gradient of Galerkin Solutions for Parabolic Problems

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