PhD Dissertation Defense - Charuka Wickramasinghe
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Charuka Wickramasinghe, PhD Candidate
A C0 FINITE ELEMENT METHOD FOR THE BIHARMONIC PROBLEM IN A POLYGONAL DOMAIN
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, and in 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. 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 refines 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.