Past Event: Oden Institute Seminar

Construction of Sparse Preconditioners for High-Order Finite Element Problems

Chetan Jhurani, Tech-X Corporation

Abstract

High-order finite elements provide better accuracy per degree of freedom than the canonical low-order finite elements. However, their implementation uses more computer memory because of denser element stiffness matrices, which in turn also leads to longer run-times when solving such systems with preconditioned methods. To reduce the run-times, researchers have considered preconditioners based on matrices assembled from sparser element stiffness matrices built from low-order elements on a separate and finer discretization [1,2]. This leads to higher accuracy due to high-order basis functions while avoiding the high cost associated due to higher memory usage. In a similar vein, we introduce a fast algebraic method of constructing sparse preconditioners for high-order finite element problems. This method creates preconditioners on individual elements by solving a constrained quadratic optimization problem for non-zero entries of the preconditioner matrix. Assembly of these local preconditioner matrices results in the global preconditioner. The sparsity pattern is chosen automatically in an algebraic manner and does not use any geometric information. Our earlier work on this topic used the eigenvalue decomposition of element stiffness matrix [3]. We have now generalized the method and made it faster so that the eigenvalue decomposition is not needed and the method works for arbitrary matrices. We present numerical results on multiple mesh types that show that using these preconditioners results in lower run-times and less memory requirements. This is a joint work with Travis Austin and Ben Jamroz at Tech-X Corporation and Marian Brezina, Tom Manteuffel, and John Ruge at University of Colorado, Boulder. [1] S. ORSZAG. Spectral methods for problems in complex geometries. J. Comp. Phys. 37 (1980), pp. 70-92. [2] J. HEYS, T. MANTEUFFEL, S. MCCORMICK, L. OLSON. Algebraic multigrid for higher-order finite elements. J. Comp. Phys. 204 (2005), pp. 520-532. [3] T. AUSTIN, M. BREZINA, T. MANTEUFFEL, J. RUGE. Automatic construction of sparse preconditioners for high-order finite element problems. Accepted for publication in Bentham eBook: Efficient Preconditioned Solution Methods for Elliptic Partial Differential Equations.