Boundary penalty finite element methods for blending surfaces — II: Biharmonic equations

In this paper the biharmonic equations are discussed, and the boundary penalty finite methods (BP-FEMs) using piecewise cubic Hermite elements are chosen to seek their approximate solutions, satisfying the normal derivative and periodical boundary conditions. Theoretical analysis is made to discover that when the penalty power σ=2,3 (or 4) and 0<σ⩽1.5 in the BP-FEM, optimal convergence rate, superconvergence and optimal numerical stability can be attained, respectively. Moreover, the normal derivative and periodical boundary conditions of the numerical solutions may even have the high convergence rates: O(h6)–O(h8), where h is the maximal boundary length of rectangular elements. A transformation for the nodal variables used is given to improve numerical stability significantly. To compromise accuracy and stability, σ=2–3 is suggested. By the techniques proposed in this paper, the elements may not be necessarily chosen to be small due to very high convergence rates.