An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes

Chunmei Wang, Junping Wang

Research output: Contribution to journalArticle

46 Scopus citations

Abstract

This paper presents a new and efficient numerical algorithm for the biharmonic equation by using weak Galerkin (WG) finite element methods. The WG finite element scheme is based on a variational form of the biharmonic equation that is equivalent to the usual H2-semi norm. Weak partial derivatives and their approximations, called discrete weak partial derivatives, are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra. The discrete weak partial derivatives serve as building blocks for the WG finite element method. The resulting matrix from the WG method is symmetric, positive definite, and parameter free. An error estimate of optimal order is derived in an H2-equivalent norm for the WG finite element solutions. Error estimates in the usual L2 norm are established, yielding optimal order of convergence for all the WG finite element algorithms except the one corresponding to the lowest order (i.e., piecewise quadratic elements). Some numerical experiments are presented to illustrate the efficiency and accuracy of the numerical scheme.

Original languageEnglish
Pages (from-to)2314-2330
Number of pages17
JournalComputers and Mathematics with Applications
Volume68
Issue number12
DOIs
StatePublished - Dec 1 2014

Keywords

  • Biharmonic equation
  • Finite element methods
  • Polyhedral meshes
  • Weak Galerkin
  • Weak partial derivatives

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