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.

翻译：我们描述了一种适用于二维自适应细化笛卡尔网格层次结构的椭圆型偏微分方程快速直接求解器。该求解器受Gillman与Martinsson提出的层次化Poincaré-Steklov（HPS）方法（SIAM J. Sci. Comput., 2014）启发，利用存储在四叉树叶节点中的局部均匀笛卡尔块上的快速求解器，是首个直接适用于由网格管理库\pforest（C. Burstedde, L. Wilcox, O. Ghattas, SIAM J. Sci. Comput. 2011）管理的自适应四叉树网格的此类求解器。在每个存储于四叉树叶节点的笛卡尔块内，我们采用基于单元中心网格的二阶有限体积离散格式。本算法的主要贡献包括：针对HPS构建阶段和解算阶段的4合1合并与分裂实现。我们通过泊松方程和亥姆霍兹方程的算例验证了求解器性能，其中网格根据右端项的高局部曲率进行自适应调整。