Constrained Finite Element Methods for Biharmonic Problem

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