Zhiqiang Cai

Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients

We propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients. The main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution. This is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions. As a consequence, the base functions are adapted to the local properties of the differential operator. In this paper, we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain a useful asymptotic solution structure. The issue of boundary conditions for the base functions will be discussed. Our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution, independently of the small scale in the homogenization limit. Application of our method to problems with continuous scales is also considered.

First-order system least squares for second-order partial differential equations: part I

This paper develops a least-squares functional that arises from recasting general second-order uniformly elliptic partial differential equations in

On the finite volume element method

n=2

The finite volume element method for diffusion equations on general triangulations

or

First-Order System Least Squares for the Stokes Equations, with Application to Linear Elasticity

3

On the accuracy of the finite volume element method for diffusion equations on composite grids

dimensions as a system of first-order equations. In part I [Z. Cai, R. D. Lazarov, T. Manteuffel, and S. McCormick, SIAM J. Numer. Anal., 31 (1994), pp. 1785--1799] a similar functional was developed and shown to be elliptic in the

Control-volume mixed finite element methods

H(\divv) \times H^1

Analysis of Velocity-Flux First-Order System Least-Squares Principles for the Navier--Stokes Equations: Part I

norm and to yield optimal convergence for finite element subspaces of

A Finite Element Method Using Singular Functions for the Poisson Equation: Corner Singularities

H(\divv) \times H^1

LEAST-SQUARES METHODS FOR LINEAR ELASTICITY ∗

. In this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first-order system. The resulting functional is proved to be elliptic in the