Grid Modification for Second Order Hyperbolic Problems

A family of Galerkin nite element methods is presented to accurately and e ciently solve the wave equation that includes sharp propagating wave fronts. The new methodology involves di erent nite element discretizations at di erent time levels; thus, at any time level, relatively coarse grids can be applied in regions where the solution changes smoothly while ner grids can be employed near wave fronts. The change of grid from time step to time step need not be continuous, and the number of grid points at di erent time levels can be arbitrarily di erent. The formulation is applicable to general second order hyperbolic equations. Stability results are proved and a-priori error estimates are established for several boundary conditions. Our error estimates consist of three parts: the time nite di erence discretization error, the spatial nite element discretization error, and the error due to the projections of the approximated solution from old grids onto new grids.