A new moving mesh algorithm for the finite element solution of variational problems

The paper is devoted to the description and application of a new iterative mesh optimization algorithm for the finite element solution of variational problems set in infinite-dimensional spaces. The optimality criterion is that the mesh should be such that the variational "energy functional," evaluated at the finite element approximation, be minimized. Such a criterion has a relatively long history in the finite element literature. The chief merit of the procedure presented in this paper is that each node of the mesh, and the corresponding nodal value of the discrete approximation, are updated by solving sequentially local minimization problems with very few degrees of freedom. It is shown that this procedure reduces the energy functional monotonically, without the need to solve the global discrete problem at intermediate stages. Applications to partial differential equations are considered. Numerical results in two dimensions are obtained by incorporating the algorithm into Bank's well-known PLTMG elliptic solver.