A posteriori L2 error estimation on anisotropic tetrahedral finite element meshes

Kunert, Gerd

doi:10.1093/imanum/21.2.503

Cited by 41 publications

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“…While the companion article [8] will be concerned with the derivation of computable a posteriori error bounds, and their implementation within automatic hp-anisotropic adaptive software, the current paper will focus on a priori error estimation. To this end, the proofs of the a priori error bounds presented in this article are based on exploiting the analysis developed in [10], which assumed that the underlying computational mesh is shape-regular and that the polynomial approximation orders are isotropic, together with the anisotropic hp-approximation results presented in [6]; for related work on anisotropic approximation theory, see [1,3,4,7,15,16], for example, and the references cited therein. We also refer to the recent article [7], where the goal-oriented error analysis of the interior penalty DGFEM on anisotropic computational meshes, assuming that the polynomial degree is kept fixed, has been developed.…”

Section: Introductionmentioning

confidence: 99%

Discontinuous Galerkin methods on hp-anisotropic meshes I: a priori error analysis

Georgoulis

Hall

Houston

2007

IJCSM

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We consider the a priori error analysis of hp-version interior penalty discontinuous Galerkin methods for second-order partial differential equations with nonnegative characteristic form under weak assumptions on the mesh design and the local finite element spaces employed. In particular, we prove a priori hp-error bounds for linear target functionals of the solution, on (possibly) anisotropic computational meshes with anisotropic tensor-product polynomial basis functions. The theoretical results are illustrated by a numerical experiment.

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Section: Introductionmentioning

confidence: 99%

Discontinuous Galerkin methods on hp-anisotropic meshes I: a priori error analysis

Georgoulis

Hall

Houston

2007

IJCSM

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“…For preconditioning a Jacobi, an Yserentant [5], or a BPX [6] preconditioner can be selected. Phase V: In the last phase the error is estimated with a residual based error estimator [7]. If the error for a volume deviates within a predefined threshold value from the maximum error, it is labeled for refinement.…”

Section: Adaptive 3-dimensional Finite Element Methodsmentioning

confidence: 99%

An Adaptive, 3-Dimensional, Hexahedral Finite Element Implementation for Distributed Memory

Hippold

Meyer

Rünger

2004

Computational Science - ICCS 2004

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Abstract. Finite elements are an effective method to solve partial differential equations. However, the high computation time and memory needs, especially for 3-dimensional finite elements, restrict the usage of sequential realizations and require efficient parallel algorithms and implementations to compute real-life problems in reasonable time. Adaptivity together with parallelism can reduce execution time significantly, however may introduce additional difficulties like hanging nodes and refinement level hierarchies. This paper presents a parallel adaptive, 3-dimensional, hexahedral finite element method on distributed memory machines. It reduces communication and encapsulates communication details like actual data exchange and communication optimizations by a modular structure.

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“…In order to measure the alignment of an anisotropic mesh T h with an anisotropic function v, a so-called matching function has been proposed by Kunert [11,12]. …”

Section: Matching Functionmentioning

confidence: 99%

“…If however the anisotropic mesh is not aligned with the solution then m 1 can be arbitrarily large (cf. [13,Numerical experiment 2] or [11,Rem. 3.3]).…”

Section: Matching Functionmentioning

confidence: 99%

Zienkiewicz–Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes

Kunert

Nicaise

2003

ESAIM: M2AN

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Abstract. We consider a posteriori error estimators that can be applied to anisotropic tetrahedral finite element meshes, i.e. meshes where the aspect ratio of the elements can be arbitrarily large. Two kinds of Zienkiewicz-Zhu (ZZ) type error estimators are derived which originate from different backgrounds. In the course of the analysis, the first estimator turns out to be a special case of the second one, and both estimators can be expressed using some recovered gradient. The advantage of keeping two different analyses of the estimators is that they allow different and partially novel investigations and results. Both rigorous analytical approaches yield the equivalence of each ZZ error estimator to a known residual error estimator. Thus reliability and efficiency of the ZZ error estimation is obtained. The anisotropic discretizations require analytical tools beyond the standard isotropic methods. Particular attention is paid to the requirements on the anisotropic mesh. The analysis is complemented and confirmed by extensive numerical examples. They show that good results can be obtained for a large class of problems, demonstrated exemplary for the Poisson problem and a singularly perturbed reaction diffusion problem.Mathematics Subject Classification. 65N15, 65N30, 65N50.

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A posteriori L2 error estimation on anisotropic tetrahedral finite element meshes

Cited by 41 publications

References 0 publications

Discontinuous Galerkin methods on hp-anisotropic meshes I: a priori error analysis

Discontinuous Galerkin methods on hp-anisotropic meshes I: a priori error analysis

An Adaptive, 3-Dimensional, Hexahedral Finite Element Implementation for Distributed Memory

Zienkiewicz–Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes

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