Optimal Eigenvalue and Asymptotic Large Time Approximations Using the Moving Finite Element Method

The Moving Finite Element method for the solution of time-dependent partial dierential equations is a numerical solution scheme which allows the automatic adaption of the nite element approximation space with time, through the use of mesh relocation (r-renement). This paper analyses the asymptotic behaviour of the method for large times when it is applied to the solution of a class of self-adjoint parabolic equations in an arbitrary number of space dimensions. It is shown that the method can produce solutions which converge to a xed mesh and it is proved that such a mesh allows an optimal approximation of the slowest decaying eigenvalue and eigenfunction for the problem. Hence it is demonstrated that the Moving Finite Element method can yield an optimal solution to such parabolic problems for large times.