On the Structure of the Moving Finite-element Equations

Abstract

The morning finite-element method for evolutionary partial differential equations leads to a coupled non-linear system of ordinary differential equations in time, with a coefficien matrix A, say, for the time derivaties, We show for linear elements in any number of dimensions, A can be written in the form M^TCM, where the matrix C depends solely on the mesh geometry and the matrix M on the gradient of the section, As a simple consequence we show that A is singular only in the cases (i) element degeneracy ($∣c∣=0$) and (ii) collinearity of nodes (M not out of full rank). We give constructions for the inversion of A in all cases. In one dimension, if A is non-singular, it has a simple explicit inverse. If A is singular we replace it by reduced matrix A^*. It can be shown that every case the spectral radius of the Jacobi iteration matrix ia ½and that A or A^* can be efficiently inverted by conjugate gradient methods. Finally, we discuss the applicability of these arguments to system of equations in any number of dimensions.