An accelerated Poisson solver based on multidomain spectral   discretization
  This paper presents a numerical method for variable coefficient elliptic PDEs with mostly smooth solutions on two dimensional domains. The PDE is discretized via a multi-domain spectral collocation method of high local order (order 30 and higher have been tested and work well). Local mesh refinement results in highly accurate solutions even in the presence of local irregular behavior due to corner singularities, localized loads, etc. The system of linear equations attained upon discretization is solved using a direct (as opposed to iterative) solver with $O(N^{1.5})$ complexity for the factorization stage and $O(N \log N)$ complexity for the solve. The scheme is ideally suited for executing the elliptic solve required when parabolic problems are discretized via time-implicit techniques. In situations where the geometry remains unchanged between time-steps, very fast execution speeds are obtained since the solution operator for each implicit solve can be pre-computed. 
Tracy Babb, Adrianna Gillman, Sijia Hao, Per-Gunnar Martinsson
math.NA