Do Young Kwak

Uniform convergence of multigrid V-cycle iterations for indefinite and nonsymmetric problems

In this paper, an analysis of a multigrid method for nonsymmetric and/or indefinite elliptic problems is presented. In this multigrid method various types of smoothers may be used. One type of smoother considered is defined in terms of an associated symmetric problem and includes point and line, Jacobi, and Gauss–Seidel iterations. Smoothers based entirely on the original operator are also considered. One smoother is based on the normal form, that is, the product of the operator and its transpose. Other smoothers studied include point and line, Jacobi, and Gauss–Seidel. It is shown that the uniform estimates of [J. H. Bramble and J. E. Pasciak, Math. Comp., 60 (1993), pp. 447–471] for symmetric positive definite problems carry over to these algorithms. More precisely, the multigrid iteration for the nonsymmetric and/or indefinite problem is shown to converge at a uniform rate provided that the coarsest grid in the multilevel iteration is sufficiently fine (but not dependent on the number of multigrid levels).

A Covolume Method Based on Rotated Bilinears for the Generalized Stokes Problem

We introduce a covolume or marker and cell (MAC) method for approximating the generalized Stokes problem on an axiparallel domain. Two grids are needed, the primal grid made up of rectangles and the dual grid of quadrilaterals. The velocity is approximated by nonconforming rotated bilinear elements with degrees of freedom at midpoints of rectangular elements and the pressure by piecewise constants. The error in the velocity in the

Optimal convergence analysis of an immersed interface finite element method

H^1_h

Accuracy and Convergence Properties of the Finite Difference Multigrid Solution of an Optimal Control Optimality System

norm and the pressure in the L2 norm are of first order, provided that the exact velocity is in H2 and the exact pressure in H1.

Mixed Covolume Methods on Rectangular Grids For Elliptic Problems

We analyze an immersed interface finite element method based on linear polynomials on noninterface triangular elements and piecewise linear polynomials on interface triangular elements. The flux jump condition is weakly enforced on the smooth interface. Optimal error estimates are derived in the broken H1-norm and L2-norm.

Mixed Covolume Methods for Elliptic Problems on Triangular Grids

The finite difference multigrid solution of an optimal control problem associated with an elliptic equation is considered. Stability of the finite difference optimality system and optimal-order error estimates in the discrete L2 norm and in the discrete H1 norm under minimum smoothness requirements on the exact solution are proved. Sharp convergence factor estimates of the two grid method for the optimality system are obtained by means of local Fourier analysis. A multigrid convergence theory is provided which guarantees convergence of the multigrid process towards weak solutions of the optimality system.

A General Framework for Constructing and Analyzing Mixed Finite Volume Methods on Quadrilateral Grids: The Overlapping Covolume Case

We consider a covolume method for a system of first order PDEs resulting from the mixed formulation of the variable-coefficient-matrix Poisson equation with the Neumann boundary condition. The system may be used to represent the Darcy law and the mass conservation law in anisotropic porous media flow. The velocity and pressure are approximated by the lowest order Raviart--Thomas space on rectangles. The method was introduced by Russell [Rigorous Block-centered Discretizations on Irregular Grids: Improved Simulation of Complex Reservoir Systems, Reservoir Simulation Research Corporation, Denver, CO, 1995] as a control-volume mixed method and has been extensively tested by Jones [A Mixed Finite Volume Elementary Method for Accurate Computation of Fluid Velocities in Porous Media, University of Colorado at Denver, 1995] and Cai et al. [Computational Geosciences, 1 (1997), pp. 289--345]. We reformulate it as a covolume method and prove its first order optimal rate of convergence for the approximate velocities as well as for the approximate pressures.

An Analysis of a Broken

We consider a covolume or finite volume method for a system of first-order PDEs resulting from the mixed formulation of the variable coefficient-matrix Poisson equation with the Neumann boundary condition. The system may represent either the Darcy law and the mass conservation law in anisotropic porous media flow, or Fourier law and energy conservation. The velocity and pressure are approximated by the lowest order Raviart--Thomas space on triangles. We prove its first-order optimal rate of convergence for the approximate velocities in the

P_1

L^2

-Nonconforming Finite Element Method for Interface Problems

-and