Constrained C0 Finite Element Methods for Biharmonic Problem

Rong An,Xuehai Huang

doi:10.1155/2012/863125

Abstract

This paper presents some constrained C0 finite element approximation methods for the biharmonic problem, which include the C0 symmetric interior penalty method, the C0 nonsymmetric interior penalty method, and the C0 nonsymmetric superpenalty method. In the finite element spaces, the C1 continuity across the interelement boundaries is obtained weakly by the constrained condition. For the C0 symmetric interior penalty method, the optimal error estimates in the broken H2 norm and in the L2 norm are derived. However, for the C0 nonsymmetric interior penalty method, the error estimate in the broken H2 norm is optimal and the error estimate in the L2 norm is suboptimal because of the lack of adjoint consistency. To obtain the optimal L2 error estimate, the C0 nonsymmetric superpenalty method is introduced and the optimal L2 error estimate is derived.

Highlights

The discontinuous Galerkin methods DGMs have become a popular method to deal with the partial differential equations, especially for nonlinear hyperbolic problem, which exists the discontinuous solution even when the data is well smooth, and the convection-dominated diffusion problem, and the advection-diffusion problem
The C1 continuity can be weakly achieved by a constrained condition that integrating the jump of the normal derivatives over the inter-element boundaries vanish
We discuss three C0 finite element methods which include the C0 symmetric interior penalty method based on the symmetric bilinear form, the C0 nonsymmetric interior penalty method, and C0 nonsymmetric superpenalty method based on the nonsymmetric bilinear forms

Summary

Introduction

The discontinuous Galerkin methods DGMs have become a popular method to deal with the partial differential equations, especially for nonlinear hyperbolic problem, which exists the discontinuous solution even when the data is well smooth, and the convection-dominated diffusion problem, and the advection-diffusion problem. Motivated by the Engel and his collaborators’ work 7 , Brenner and Sung in 8 studied the C0 interior penalty method for fourth-order problem on polygonal domains. They used the C0 finite element solution to approximate C1 solution by a postprocessing procedure, and the C1 continuity can be achieved weakly by the penalty on the jump of the normal derivatives on the interelement boundaries. The C1 continuity can be weakly achieved by a constrained condition that integrating the jump of the normal derivatives over the inter-element boundaries vanish Under this constrained condition, we discuss three C0 finite element methods which include the C0 symmetric interior penalty method based on the symmetric bilinear form, the C0 nonsymmetric interior penalty method, and C0 nonsymmetric superpenalty method based on the nonsymmetric bilinear forms. In order to improve the order of the L2 error estimate, we give the C0 nonsymmetric superpenalty method and show the optimal L2 error estimates

C0 Finite Element Approximation

C0 Symmetric Interior Penalty Method

C0 Nonsymmetric Interior Penalty Method

C0 Superpenalty Nonsymmetric Method