high performance computing on graphics processing units: hgpu.org

Space-time finite element method provides a robust and accurate alternative to the traditional FEM based on semi-discrete schemes due to its extended capability in establishing approximations in both space and time. The extended capability, however, requires the simultaneous discretization of spatial and temporal domains. This subsequently results in a system of equations that is considerably larger in size than those obtained in the standard finite element formulation. In general, solving equations generated based on the space-time formulation requires substantially more computing time. In some cases, it becomes a bottle neck for practical implementation due to the large number of degrees of freedom. There is thus a need to explore ways to accelerate the procedure for finding a solution to the system of equations in the space-time method. With the recent developments in the use of Graphics Processing Units (GPUs) for general purpose parallel computations (GPGPU), an effort is made in this thesis to explore the possibility of developing a GPU based solver for the space-time finite element method to accelerate the computation. A two-step approach is taken: In the first step, the GPU version of the direct solver based on LAPACK and a preconditioned conjugate gradient method are tested for systems of linear equations involving both dense and sparse matrices. Both methods are shown to significantly accelerate the process of finding the solution. Subsequently, the developed GPU-based algorithms are implemented on the system of linear equations obtained from space-time FEM and enriched space-time FEM. It is reported that GPU-based implementation yields significant speed up. Based on these implementations, it is concluded that the GPU-based system could serve as an effective platform for the space-time method.