Weak Galerkin Finite Element Methods for the Biharmonic Equation on Polytopal Meshes

Abstract

A new weak Galerkin (WG) finite element method is introduced and analyzed in this paper for the biharmonic equation in its primary form. This method is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on general finite element partitions consisting of polygons or polyhedra of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter-free. Optimal order error estimates in a discrete $H^2$ norm is established for the corresponding WG finite element solutions. Error estimates in the usual $L^2$ norm are also derived, yielding a sub-optimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence under suitable regularity assumptions.

23 pages, 1 figure, 2 tables. arXiv admin note: text overlap with arXiv:1202.3655, arXiv:1204.3655

References in corpus (1)

• A Weak Galerkin Finite Element Method for the Stokes Equations

Cited by in corpus (8)

• A Primal-Dual Weak Galerkin Finite Element Method for Fokker-Planck Type Equations
• A Weak Galerkin Mixed Finite Element Method for second order elliptic equations on 2D Curved Domains
• A Primal-Dual Weak Galerkin Finite Element Method for Second Order Elliptic Equations in Non-Divergence Form
• A Locking-Free Weak Galerkin Finite Element Method for Elasticity Problems in the Primal Formulation
• A Weak Galerkin Finite Element Method for the Maxwell Equations
• An Efficient Numerical Scheme for the Biharmonic Equation by Weak Galerkin Finite Element Methods on Polygonal or Polyhedral Meshes
• A Hybridized Formulation for the Weak Galerkin Mixed Finite Element Method
• A Weak Galerkin Finite Element Method for A Type of Fourth Order Problem Arising From Fluorescence Tomography