SAMS Subject Classification Number: 23

In this project we investigate techniques for solving second-order elliptic ordinary and partial differential equations with variable coefficients on one- and two-dimensional domains. We explore spectral collocation methods and domain decomposition strategies for this task, and derive a Hierarchical Poincaré–Steklov (HPS) based approach, similar to that introduced by Gillman and Martinsson [1]. The HPS method is a recursive domain decomposition. It merges solution and Dirichlet-to-Neumann operators between subdomains, enforcing continuity of the solution and its derivative across domain boundaries. The result is a spectrally accurate discretization with an explicit fast direct solve that can be applied to problems with smooth solutions on any domain which can be decomposed into rectangular subdomains. A major advantage of the HPS strategy is that after the solution and Dirichlet-to-Neumann operators have been constructed, applying the boundary conditions and solving the problem is computationally inexpensive, meaning the scheme is ideal for solving the same problem with different boundary data. An additional benefit of the approach is that it can be parallelised across multiple processors to minimize computation time. We derive and implement the spectral HPS approach, and demonstrate its efficiency on some simple examples.

[1] A. Gillman and P.G. Martinsson, A direct solver with O(N) complexity for variable coefficient elliptic PDEs discretized via a high-order composite spectral collocation method, SIAM Journal on Scientific Computing \(\mathbf{36}\) (2014), A2023-2046.