An accelerated Poisson solver based on multidomain spectral discretization
It provides an efficient direct solver for elliptic PDEs with local refinement, benefiting computational scientists solving time-dependent problems with fixed geometry.
The paper presents a direct solver for variable coefficient elliptic PDEs discretized via high-order multidomain spectral collocation, achieving O(N^{1.5}) factorization and O(N log N) solve complexity, enabling fast implicit time-stepping for parabolic problems.
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.