On steady and large time solutions of the semi-discrete Moving Finite Element equations for one-dimensional diffusion problems

This paper considers how the Moving Finite Element (MFE) methodapproxim ates the steady and large time solutions of a familyof linear diffusion equations in one space dimension. In particular,it is shown that any steady solution to the Moving Finite Elementequations must satisfy the stationary equations for a best approximationto the steady solution of the PDE from the manifold of free-knotlinear splines, in some problem dependent norm. For the special case of the inhomogeneous linear heat equationit is also shown that, under certain conditions, the only steadyMFE solution is the unique global best fit to the true steadysolution, in the H¹ semi-norm. It is also demonstrated numericallythat these steady solutions are stable attractors. Finally,a numerical study of the large time solutions of the homogeneouslinear heat equation is undertaken and it is demonstrated thatthe MFE solutions appear to possess a rather novel temporalaccuracy property.