An anisotropic mesh adaptation method for the finite element solution of variational problems

Weizhang Huang, Xianping Li

Research output: Contribution to journalArticlepeer-review

10 Scopus citations

Abstract

It has been amply demonstrated that anisotropic mesh adaptation can significantly improve computational efficiency over isotropic mesh adaptation especially for problems with strong anisotropic features. Although numerous research has been done on isotropic mesh adaptation for finite element solution of variational problems, little work has been done on anisotropic mesh adaptation. In this paper we consider anisotropic mesh adaptation method for the finite element solution of variational problems. A bound for the first variation of a general functional is derived, which is semi-a posteriori in the sense that it involves the residual and edge jump, both dependent on the computed solution, as well as the Hessian of the exact solution. A formula for the metric tensor M for use in anisotropic mesh adaptation is defined such that the bound is minimized on a mesh that is uniform in the metric specified by M (i.e., an M-uniform mesh). Interestingly, when restricted to isotropic meshes, we can obtain a similar but completely a posteriori bound and the corresponding formula for the metric tensor. When M is defined, an anisotropic adaptive mesh is generated as an M-uniform mesh. Numerical results demonstrate that the new mesh adaptation method is comparable in performance with existing ones based on interpolation error and has the advantage that the resulting mesh also adapts to changes in the structure of the underlying problem.

Original languageEnglish (US)
Pages (from-to)61-73
Number of pages13
JournalFinite Elements in Analysis and Design
Volume46
Issue number1-2
DOIs
StatePublished - Jan 2010
Externally publishedYes

Keywords

  • Anisotropic mesh adaptation
  • Finite element
  • Mesh adaptation
  • Variational problem

ASJC Scopus subject areas

  • Analysis
  • Engineering(all)
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics

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