# Comparison of Mesh and Meshless Methods for Partial Differential Equations of Galerkin Form

12-1-2013

Thesis

## Degree Name

Master of Science (MS)

## Department

Mathematical Sciences

Xin Li

Michael Marcuzzi

David Hatchett

80

## Abstract

There are two purposes of this research project. The first purpose is to compare two types of Galerkin methods: The finite element mesh method and moving least sqaures meshless Galerkin (EFG) method. The second purpose of this project is to determine if a hybrid between the mesh and meshless method is beneficial.

This manuscript will be divided into three main parts. The first part is chapter one which develops the finite element method. The second part (Chapter two) will be developing the meshless method. The last part will provide a method for combining the mesh and meshless methods for a hybrid method.

The mesh and meshfree methods will be studied from a theoretical framework. From this framework, a computational program will be implemented for mesh, meshfree and the hybrid method. The program will produce the domain and the solution of the problem graphically as well as give relative error between the exact solution and the approximate solution. This process will be repeated for many domains and PDES problems of the form Lu=f.

The mesh method (Finite Element Method) uses a uniform partition of the domain called a triangulation. Each of the triangles in the domain is called an element. A one-to-four refinement method can then be applied to each triangle to produce a uniform mesh. The elements are then used to form a matrix equation using the Galerkin form of the PDE and solve the system for the approximate solution. On the other hand, the meshfree method (EFG) will use the same uniform mesh as the Finite Element Method as well as a scatter plot for direct comparisons using rectangular support domains. This method uses the geometry at a point instead of an element. Similar to the Finite Element method, Moving least squares uses a linear algebraic system to solve for the approximate solution.

Finally a hybrid method will first do an error analysis between each method. The error analysis determines what methods will be combined.

## Keywords

Finite element method; Galerkin methods; Meshfree methods (Numerical analysis)

## Disciplines

Applied Mathematics | Mathematics | Numerical Analysis and Computation

pdf

English