A C0 Finite Element Method For The Biharmonic Problem In A Polygonal Domain

Abstract

This dissertation studies the biharmonic equation with Dirichlet boundary conditions in a polygonal domain. The biharmonic problem appears in various real-world applications, for example in plate problems, human face recognition, radar imaging, and hydrodynamics problems. There are three classical approaches to discretizing the biharmonic equation in the literature: conforming finite element methods, nonconforming finite element methods, and mixed finite element methods. We propose a mixed finite element method that effectively decouples the fourth-order problem into a system of one steady-state Stokes equation and one Poisson equation. As a generalization to the above-decoupled formulation, we propose another decoupled formulation using a system of two Poison equations and one steady-state Stokes equation. We solve Poisson equations using linear and quadratic Lagrange\u27s elements and the Stokes equation using Hood-Taylor elements and Mini finite elements. It is shown that the solution of each system is equivalent to that of the original fourth-order problem on both convex and non-convex polygonal domains. Two finite element algorithms are, in turn, proposed to solve the decoupled systems. Solving this problem in a non-convex domain is challenging due to the singularity occurring near re-entrant corners. We introduce a weighted Sobolev space and a graded mesh refine Algorithm to attack the singularity near re-entrant corners. We show the regularity results of each decoupled system in both Sobolev space and weighted Sobolev space. We derive the $H^1$ and $L^2$ error estimates for the numerical solutions on quasi-uniform and graded meshes. We present various numerical test results to justify the theoretical findings. Given the availability of fast Poisson solvers and Stokes solvers, our Algorithm is a relatively easy and cost-effective alternative to existing algorithms for solving the biharmonic equation