Date of Award

4-28-2023

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

First Advisor

Xiu Ye

Abstract

The weak Galerkin finite element method is a flexible and effective numerical technique for solving partial differential equations (PDEs). The goal of this dissertation is to develop new weak Galerkin schemes for solving PDEs on uniform triangular partitions in 2D. These schemes will improve the efficiency of the weak Galerkin method and avoiding numerical locking. First, we investigate the lowest-order weak Galerkin finite element method for solving reaction-diffusion equations with singular perturbations. Then, we propose a weak Galerkin method for the Laplace equation using harmonic polynomial finite elements instead of the full polynomial space $P_k$ to achieve the same order of accuracy and convergence, thus improving the efficiency. Moreover, we present and analyze a stabilizer free weak Galerkin finite element method for the second-order parabolic partial differential equation with initial boundary conditions. Furthermore, we also develop numerical methods for solving time-dependent convection-diffusion equations with initial-boundary conditions using a stabilizer free weak Galerkin finite element method. The optimal convergence rate is derived in both $H^1$ and $L^2$ norms for the corresponding weak Galerkin approximation and stabilizer free weak Galerkin approximation. The numerical examples are presented to validate the theoretical conclusions.

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