Access Type

Open Access Dissertation

Date of Award

January 2022

Degree Type

Dissertation

Degree Name

Ph.D.

Department

Mathematics

First Advisor

Hengguang Li

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's 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.

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