| A study of the Galerkin finite element method on higher order partial differential equations was performed with the long term goal of achieving a numerical solution methodology for the nonlinear partial differential equation called the Sawada Kotera equation, whose highest order derivative is fifth order in space. In this thesis, the goal was to find numerical solutions to linear partial differential equations with third, fourth, and fifth order derivatives, as a precursor towards a general technique for nonlinear partial differential equations such as the Sawada Kotera equation. In this study, we determined that the Galerkin finite element method produced equations that did not have enough degrees of freedom to correctly discretize the derivatives with order three or higher. In order to fix these problems, the average Galerkin finite element method was introduced. The average Galerkin finite element method is accomplished by averaging the local matrix of a derivative at each node rather than using it once per element. The average Galerkin finite element method achieved the same or better order of accuracy than the Galerkin finite element method for partial differential equations with first and second order derivatives. Using the average Galerkin finite element method with quadratic basis functions, we were able to properly calculate third and fourth order derivatives. Using the average Galerkin finite element method with cubic basis functions, we were able to properly calculate fifth order derivatives. |
