Date of Award

Summer 8-2018

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Committee Chair

Huiqing Zhu

Committee Chair Department

Mathematics

Committee Member 2

C.S. Chen

Committee Member 2 Department

Mathematics

Committee Member 3

Haiyan Tian

Committee Member 3 Department

Mathematics

Committee Member 4

Zhifu Xie

Committee Member 4 Department

Mathematics

Abstract

There are many types of adaptive methods that have been developed with different algorithm schemes and definitions for solving Partial Differential Equations (PDE). Adaptive methods have been developed in mesh-based methods, and in recent years, they have been extended by using meshfree methods, such as the Radial Basis Function (RBF) collocation method and the Method of Fundamental Solutions (MFS). The purpose of this dissertation is to introduce an adaptive algorithm with a residual type of error estimator which has not been found in the literature for the adaptive MFS. Some modifications have been made in developing the algorithm schemes depending on the governing equations, the domains, and the boundary conditions. The MFS is used as the main meshfree method to solve the Laplace equation in this dissertation, and we propose adaptive algorithms in different versions based on the residual type of an error estimator in 2D and 3D domains. Popular techniques for handling parameters and different approaches are considered in each example to obtain satisfactory results. Dirichlet boundary conditions are carefully chosen to validate the efficiency of the adaptive method. The RBF collocation method and the Method of Approximate Particular Solutions (MAPS) are used for solving the Poisson equation. Due to the type of the PDE, different strategies for constructing the adaptive method had to be followed, and proper error estimators are considered for this part. This results in having a new point of view when observing the numerical results. Methodologies of meshfree methods that are employed in this dissertation are introduced, and numerical examples are presented with various boundary conditions to show how the adaptive method performs. We can observe the benefit of using the adaptive method and the improved error estimators provide better results in the experiments.

ORCID ID

0000-0003-3760-664X

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