Numerical solutions of biharmonic equations on non-convex polygonal domains

Abstract

To construct C1-continuous basis functions for the numerical solutions of two dimensional biharmonic equations on non-convex domains with clamped and/or simply supported boundary conditions, we use B-spline basis functions instead of conventional Hermite finite element basis functions. The C1-continuous B-spline approximation functions constructed on the master patch are moved onto a physical domain by a geometric patch mapping. However it is difficult to cover a non-convex polygonal domain by one smooth patch mapping. Hence we decompose a non-convex domain into several overlapping rectangular subdomains and use the Schwartz alternating method to assemble local solutions for the global solution. Furthermore, in order to handle the corner singularities arising in non-convex domains, we introduce the implicitly enriched Galerkin method, which is similar to the explicit enrichment techniques used in XFEM, G-FEM, PUFEM, that are successful in handling the singularity problems arising in second-order differential problems.