Constrained C 0 Finite Element Methods for Biharmonic Problem

Abstract

This paper presents some constrained $C^0$ finite element approximation methods for the biharmonic problem, which include the $C^0$ symmetric interior penalty method, the $C^0$ nonsymmetric interior penalty method, and the $C^0$ nonsymmetric superpenalty method. In the finite element spaces, the $C^1$ continuity across the interelement boundaries is obtained weakly by the constrained condition. For the $C^0$ symmetric interior penalty method, the optimal error estimates in the broken $H^2$ norm and in the $L^2$ norm are derived. However, for the $C^0$ nonsymmetric interior penalty method, the error estimate in the broken $H^2$ norm is optimal and the error estimate in the $L^2$ norm is suboptimal because of the lack of adjoint consistency. To obtain the optimal $L^2$ error estimate, the $C^0$ nonsymmetric superpenalty method is introduced and the optimal $L^2$ error estimate is derived.