A Parallel Implementation of the Galerkin Method for Solving Partial Differential Equations on a Triangular Mesh

Abstract

Finite Element Methods are techniques for estimating solutions to boundary value problems for partial differential equations from an approximating subspace. These methods are based on weak or variational forms of the BVP that require less of the problem functions than what the original PDE would suggest in terms of order of differentiability and continuity. In the scope of this project, we focused on implementing the Galerkin Finite Element Method, which provides a best approximation to the true solution from a finite-dimensional subspace of piecewise polynomial functions defined on a triangular mesh. For this thesis, we developed a shared memory parallel implementation of the Galerkin Method that can be executed on a GPU to minimize runtime by means of multiple processors working simultaneously in unison on each calculation. For this purpose, we used the open-source libraries PyOpenCL and Loo.py. Thus we are able to explore how essential tasks in the solution process map onto shared memory platforms, such as the construction of the stiffness matrix from the connectivity data of the triangular mesh that may then be used to approximate the true solution with numerical methods.