Parallel h-p Spectral Element Methods for Elliptic Problems on Non-smooth Domains

We propose a new parallel h-p Spectral element method to solve elliptic boundary value problems with mixed Neumann and Dirichlet boundary conditions on non-smooth domains. The method is shown to be exponentially accurate and asymptotically faster than the standard h-p finite element method. We use the auxiliary mapping of the form of z = ln ξ. The spectral element functions we use are fully non-conforming for pure Dirichlet problems. However, for mixed problems we need to make it conforming only at the vertices of the elements. The dimension of the resulting Schur complement matrix is quite small and this enables us to construct an accurate approximation of the Schur system. The method is a least-squares collocation method and the resulting normal equations are solved using preconditioned conjugate gradient method. We state the differentiability, stability and error estimates and discuss the numerical scheme and parallelization techniques. Using the stability estimates a parallel preconditioner with optimal condition number (polylogarithmic) is obtained. The algorithm is suitable for a distributed memory parallel computer. Load balancing issues are discussed and inter-processor communication involved is shown to be small. Finally, we provide the numerical results of a number of model problems which confirm the theoretical estimates and demonstrate the robustness of the method.