14 research outputs found
Compression and Reduced Representation Techniques for Patch-Based Relaxation
Patch-based relaxation refers to a family of methods for solving linear
systems which partitions the matrix into smaller pieces often corresponding to
groups of adjacent degrees of freedom residing within patches of the
computational domain. The two most common families of patch-based methods are
block-Jacobi and Schwarz methods, where the former typically corresponds to
non-overlapping domains and the later implies some overlap. We focus on cases
where each patch consists of the degrees of freedom within a finite element
method mesh cell. Patch methods often capture complex local physics much more
effectively than simpler point-smoothers such as Jacobi; however, forming,
inverting, and applying each patch can be prohibitively expensive in terms of
both storage and computation time. To this end, we propose several approaches
for performing analysis on these patches and constructing a reduced
representation. The compression techniques rely on either matrix norm
comparisons or unsupervised learning via a clustering approach. We illustrate
how it is frequently possible to retain/factor less than 5% of all patches and
still develop a method that converges with the same number of iterations or
slightly more than when all patches are stored/factored.Comment: 16 pages, 5 figure
Numerical Simulation of Multiphase Flow in Fractured Porous Media
Fractures provide preferred paths for flow and transport in many porous media. They have a significant influence on process behavior in groundwater remediation, reservoir engineering and safety analysis for waste repositories. We present a finite volume method for the numerical solution of the multiphase flow equations in fractured porous media. The capillary pressure is treated by an extended capillary pressure interface condition. The method is fully coupled and fully implicit and employs a mixed-dimensional formulation with lower dimensional elements in the fractures. The method features unstructured grids, adaptive refinement and multigrid methods. It is implemented for twodimensional and threedimensional complex problems with several million unknowns. Additionally, a discontinuous Galerkin method for the groundwater flow equation and its multigrid treatment is presented
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described