34 research outputs found
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Final Report on subcontract B551021: Optimal AMG interpolation and Convergence theory
The goal of this project is to implement and study various techniques for the construction of Algebraic Multigrid Methods (AMG) for the solution of positive definite linear systems arising from the discretizations of elliptic partial differential equations (PDEs). Both theoretical as well as practical implementation of the methods that we have developed are based on compatible relaxation and energy minimization
A simple preconditioner for a discontinuous Galerkin method for the Stokes problem
In this paper we construct Discontinuous Galerkin approximations of the
Stokes problem where the velocity field is H(div)-conforming. This implies that
the velocity solution is divergence-free in the whole domain. This property can
be exploited to design a simple and effective preconditioner for the final
linear system.Comment: 27 pages, 4 figure
Final Report on Subcontract B591217: Multigrid Methods for Systems of PDEs
Progress is summarized in the following areas of study: (1) Compatible relaxation; (2) Improving aggregation-based MG solver performance - variable cycle; (3) First Order System Least Squares (FOSLS) for LQCD; (4) Auxiliary space preconditioners; (5) Bootstrap algebraic multigrid; and (6) Practical applications of AMG and fast auxiliary space preconditioners
First-Order System Least Squares and the Energetic Variational Approach for Two-Phase Flow
This paper develops a first-order system least-squares (FOSLS) formulation
for equations of two-phase flow. The main goal is to show that this
discretization, along with numerical techniques such as nested iteration,
algebraic multigrid, and adaptive local refinement, can be used to solve these
types of complex fluid flow problems. In addition, from an energetic
variational approach, it can be shown that an important quantity to preserve in
a given simulation is the energy law. We discuss the energy law and inherent
structure for two-phase flow using the Allen-Cahn interface model and indicate
how it is related to other complex fluid models, such as magnetohydrodynamics.
Finally, we show that, using the FOSLS framework, one can still satisfy the
appropriate energy law globally while using well-known numerical techniques.Comment: 22 pages, 8 figures submitted to Journal of Computational Physic
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Extending the applicability of multigrid methods
Multigrid methods are ideal for solving the increasingly large-scale problems that arise in numerical simulations of physical phenomena because of their potential for computational costs and memory requirements that scale linearly with the degrees of freedom. Unfortunately, they have been historically limited by their applicability to elliptic-type problems and the need for special handling in their implementation. In this paper, we present an overview of several recent theoretical and algorithmic advances made by the TOPS multigrid partners and their collaborators in extending applicability of multigrid methods. Specific examples that are presented include quantum chromodynamics, radiation transport, and electromagnetics