An adaptive high order direct solution technique for elliptic boundary value problems

TL;DR

This paper introduces an adaptive high order discretization method for elliptic boundary value problems, enhancing the Hierarchical Poincaré-Steklov approach with a more efficient basis and an adaptive refinement criterion.

Contribution

It presents a modified tensor product basis and an adaptive refinement technique that improve efficiency, stability, and accuracy of the HPS method for elliptic problems.

Findings

Achieves prescribed accuracy with adaptive refinement.

Provides an efficient direct solver for elliptic problems.

Extends applicability to time-dependent problems.

Abstract

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 Poincar\'e-Steklov (HPS) method. Roughly speaking, the HPS method is based on local pseudospectral discretizations glued together with Poincar\'e-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.