On the Mass Matrix Spectrum Bounds of Wathen and the Local Moving Finite Elements of Baines

Abstract

Andrew Wathen has shown that the eigenvalues of the diagonally-preconditioned piecewise linear moving finite element (MFE) or finite element (FE) mass matrix in n dimensions lie in $[\frac{1}{2},1 + \frac{1}{2}n]$. Baines, using similar considerations, has designed “very local” MFE methods with block-diagonal mass matrices. In this paper a simplified proof of Wathen’s basic spectrum bound is given. It results from a simple comparison bound between the $L^2 $ norm and a certain “diagonal norm” for discontinuous piecewise linear functions on an arbitrary triangular grid. The simplicity of our comparison lets us extend Wathen’s result to other situations such as Miller’s gradient-weighted MFE method (GWMFE). It also lets us design “very local” MFE (and FE and GWMFE) methods which minimize the PDE residual $\dot u - L(u)$ not in $L^2 $ norm but in a comparable norm. These methods turn out to be equivalent to those of Baines. These methods retain the desired conservation properties for PDE’s in “conservation l...