Optimal eigenvalue and asymtotic large-time approximations using the moving finite-element method

The moving finite-element method for the solution of time-dependent partial differential equations is a numerical solution scheme which allows the automatic adaptation of the finite-element approximation space with time, through the use of mesh relocation (r-refinement). This paper analyzes 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 asymptotically the method will produce solutions which converge to a fixed 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.