A best approximation property of the moving finite element method

The moving finite element method (MFE) for the solution of time-dependent partial differential equations (PDEs) is a numerical solution scheme which allows the automatic adaption of the finite element approximation space with time. An analysis of how this method models the steady solutions of a general class of parabolic linear source equations is presented.

It is shown that under certain conditions the steady solutions of the MFE problem can correspond to best free knot spline approximations to the true steady solution of the differential equation when using the natural norm associated with the problem. Hence a quantitative measure of the advantages of the MFE method over the usual fixed grid Galerkin method is produced for these equations. A number of numerical examples are included to illustrate these results.