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Dissertation Information
Title: An Adaptive And Parallel Direct Solver For Elliptic Partial Differential Equations
Program: Doctor of Philosophy in Computing
Advisor: Dr. Donna Calhoun, Mathematics
Committee Members: Dr. Michal Kopera, Mathematics and Dr. Grady Wright, Mathematics
Abstract
An adaptive extension of the Hierarchical Poincaré-Steklov (HPS) method (Gillman & Martinsson, 2014) is motivated, derived, and analyzed for use in solving elliptic partial differential equations (PDE). The quadtree-adaptive HPS (QAHPS) method is a direct method for use with adaptive mesh refinement (AMR) techniques for solving elliptic PDEs on a hierarchy of adaptively refined finite volume meshes. The QAHPS method builds up a solution operator set in O(N^(3/2)) time that acts as the factorization of the system matrix, with the expected linear O(N) in time for the application of the solution operator set to any number of right-hand side vectors. The solution operator set can be adapted efficiently as the mesh discretization changes, providing a novel adaptive direct solver. The QAHPS method is outlined for both serial and parallel computer architectures, with detailed implementation details for use with the mesh library p4est (Burstedde et al., 2011). Strong and weak scaling results are shown from runs on the Polaris supercomputer. Applications of the QAHPS method to the Laplace, Poisson, and Helmholtz equations are provided, with comparisons to other HPS implementations and other solvers for elliptic PDEs.