Constrained \(C^0\) finite element methods for biharmonic problem (Q1938315)

Summary: 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.