Summary: We propose a new \(h\)-\(p\) spectral element method to solve elliptic boundary value problems with mixed Neumann and Dirichlet boundary conditions on nonsmooth domains. The method is shown to be exponentially accurate and asymptotically faster than the standard \(h\)-\(p\) finite element method. The spectral element functions are fully non-conforming for pure Dirichlet problems and conforming only at the vertices of the elements for mixed problems, and hence, the dimension of the resulting Schur complement matrix is quite small.
The method is a least-squares collocation method and the resulting normal equations are solved using preconditioned conjugate gradient method with an almost optimal preconditioner. The algorithm is suitable for a distributed memory parallel computer. The numerical results of a number of model problems are presented, which confirm the theoretical estimates.