On the Mass Matrix Spectrum Bounds of Wathen and the Local Moving Finite Elements of Baines
Keith Miller · SIAM Journal on Numerical Analysis · 1992
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 law” form. Finally, we discuss the possibilities for “combined explicit-implicit” codes for certain problems with shocks.