An adaptive high order direct solution technique for elliptic boundary value problems
For computational scientists solving elliptic PDEs, this provides a more efficient and stable adaptive solver that enables implicit time stepping in time-dependent problems.
This work presents an adaptive high-order discretization for elliptic boundary value problems using an updated Hierarchical Poincaré-Steklov method with a more efficient and stable tensor product basis. The technique achieves user-prescribed accuracy and enables efficient direct solvers, extending applicability to time-dependent problems.
This manuscript presents an adaptive high order discretization technique for elliptic boundary value problems. The technique is applied to an updated version of the Hierarchical Poincaré-Steklov (HPS) method. Roughly speaking, the HPS method is based on local pseudospectral discretizations glued together with Poincaré-Steklov operators. The new version uses a modified tensor product basis which is more efficient and stable than previous versions. The adaptive technique exploits the tensor product nature of the basis functions to create a criterion for determining which parts of the domain require additional refinement. The resulting discretization achieves the user prescribed accuracy and comes with an efficient direct solver. The direct solver increases the range of applicability to time dependent problems where the cost of solving elliptic problems previously limited the use of implicit time stepping schemes.