So-Hsiang Chou

Error estimates in L 2 , H 1 and L ∞ in covolume methods for elliptic and parabolic p roblems: a unified approach

In this paper we consider covolume or finite volume element methods for variable coefficient elliptic and parabolic problems on convex smooth domains in the plane. We introduce a general approach for connecting these methods with finite element method analysis. This unified approach is used to prove known convergence results in the H 1 , L 2 norms and new results in the max-norm. For the elliptic problems we demonstrate that the error u-u h between the exact solution u and the approximate solution u h in the maximum norm is O(h 2 |ln h|) in the linear element case. Furthermore, the maximur norm error in the gradient is shown to be of first order. Similar results hold for the parabolic problems.

Analysis and convergence of a covolume method for the generalized Stokes problem

We introduce a covolume or MAC-like method for approximating the generalized Stokes problem. Two grids are needed in the discretization; a triangular one for the contimlity equation and a quadrilateral one for the momentum equation. The velocity is approximated using nonconforming piecewise linears and the pressure piecewise constants. Error in the L 2 norm for the pressure and error in a mesh dependent H 1 norm as well as in the L 2 norm for the velocity are shown to be of first order, provided that the exact velocity is in H 2 and the true pressure in H I . We also introduce the concept of a network model into the discretized linear system so that an efficient pressure-recovering technique can be used to simplify a great deal the computational work involved in the augmented Lagrangian method. Given is a very general decomposition condition under which this technique is applicable to other fluid problems that can be formulated as a saddle-point problem.

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

A general mixed covolume framework for constructing conservative schemes for elliptic problems

H^1_h

Optimal convergence analysis of an immersed interface finite element method

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 present a general framework for the finite volume or covolume schemes developed for second order elliptic problems in mixed form, i.e., written as first order systems. We connect these schemes to standard mixed finite element methods via a one-to-one transfer operator between trial and test spaces. In the nonsymmetric case (convection-diffusion equation) we show one-half order convergence rate for the flux variable which is approximated either by the lowest order Raviart-Thomas space or by its image in the space of discontinuous piecewise constants. In the symmetric case (diffusion equation) a first order convergence rate is obtained for both the state variable (e.g., concentration) and its flux. Numerical experiments are included.

Mixed Covolume Methods for Elliptic Problems on Triangular Grids

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.

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.

Analysis and Convergence of a MAC-like Scheme for the Generalized Stokes Problem

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

Mixed finite volume methods on nonstaggered quadrilateral grids for elliptic problems

L^2