Global superconvergence for blending surfaces by boundary penalty plus hybrid FEMs

Abstract

In this paper, consider biharmonic equations and 3D blending surfaces, and choose the bi-cubic Hermite elements to seek their approximate solutions. We pursue not only the global superconvergence originated in Lin and his colleagues (Lin, 1994; Lin and Luo, 1995; Lin and Yan, 1996), but also better numerical stability. The boundary penalty plus hybrid integrals are employed to satisfy the normal derivative boundary conditions. Compared with the penalty finite methods (BP-FEM) of bi-cubic Hermite elements in (Li, 1998, 1999; Li and Chang, 1999), the merit of the new methods in this paper is reduction of σ's values thus to improve numerical stability. Suppose that the solution domain Ω can be split into small rectangles □ ij . The global superconvergence O( h 2.5) and O( h 3.5) in H 2 norms are achieved for quasiuniform and uniform □ ij , respectively. Both cases yield the optimal condition number O( h −4), compared with O( h −6) and O( h −8) in (Li, 1999; Li and Chang, 1999). This is an important improvement of stability for biharmonic solutions. However, for 3D blending surfaces, only the global superconvergence O( h 3) is achieved for uniform □ ij . Morever, numerical experiments are provided for biharmonic equations to support the high superconvergence O( h 3.5) involving the natural boundary condition for uniform □ ij with parameter μ∈[0,1). This paper manifests a great flexibility of global superconvegence in applications.