A Fast Direct Solver for Elliptic PDEs on a Hierarchy of Adaptively Refined Quadtrees

TL;DR

This paper introduces a fast, direct solver for elliptic PDEs on adaptively refined 2D quadtree meshes, combining hierarchical Poincaré-Steklov methods with local finite volume discretizations for efficient computation.

Contribution

It presents the first adaptive quadtree-based direct solver for elliptic PDEs that integrates HPS methods with a grid management library, enabling efficient solutions on locally refined meshes.

Findings

Solver efficiently handles Poisson and Helmholtz problems.

Demonstrates effectiveness on meshes adapted to high curvature regions.

Achieves fast build and solve stages with adaptive refinement.

Abstract

We describe a fast, direct solver for elliptic partial differential equations on a two-dimensional hierarchy of adaptively refined, Cartesian meshes. Our solver, inspired by the Hierarchical Poincar\'e-Steklov (HPS) method introduced by Gillman and Martinsson (SIAM J. Sci. Comput., 2014) uses fast solvers on locally uniform Cartesian patches stored in the leaves of a quadtree and is the first such solver that works directly with the adaptive quadtree mesh managed using the grid management library \pforest (C. Burstedde, L. Wilcox, O. Ghattas, SIAM J. Sci. Comput. 2011). Within each Cartesian patch, stored in leaves of the quadtree, we use a second order finite volume discretization on cell-centered meshes. Key contributions of our algorithm include 4-to-1 merge and split implementations for the HPS build stage and solve stage, respectively. We demonstrate our solver on Poisson and Helmholtz problems with a mesh adapted to the high local curvature of the right-hand side.