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

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

A simple uniformly convergent iterative method for the non-symmetric incomplete interior penalty discontinuous Galerkin discretization

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Uniformly convergent iterative methods for discontinuous Galerkin discretizations

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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

AN EXPONENTIAL FITTING SCHEME FOR GENERAL CONVECTION-DIFFUSION EQUATIONS ON TETRAHEDRAL MESHES

Abstract. This paper contains construction and analysis a finite element approximation for convection dominated diffusion problems with full coefficient matrix on general simplicial partitions in Ê d, d ≥ 2. This construction is quite close to the scheme of Xu and Zikatanov [22] where a diagonal coefficient matrix has been considered. The scheme is of the class of exponentially fitted methods that does not use upwind or checking the flow direction. It is stable for sufficiently small discretization step-size assuming that the boundary value problem for the convection-diffusion equation is uniquely solvable. Further, it is shown that, under certain conditions on the mesh the scheme is monotone. Convergence of first order is derived under minimal smoothness of the solution. 1