R-REFINEMENT FOR EVOLUTIONARY PDES WITH FINITE-ELEMENTS OR FINITE-DIFFERENCES

In this paper two different moving-mesh methods (r-refinement) are app lied to evolutionary PDE models in one and two space dimensions. The f irst method (moving finite elements) is based on a minimization of the PDE residual that is obtained by approximating the solution with piec ewise linear elements. The second method (moving finite differences) i s based on an equidistribution principle with smoothing both in the sp atial and the temporal direction. Theory predicts that the finite-elem ent based moving-mesh method moves its grid points with the flow of th e PDE, whereas the finite-difference based method moves its grid point s with the steep parts of the PDE solution, respectively. Numerical ex periments show some differences and similarities between the finite-el ement and finite-difference case when applied to 1D and 2D time-depend ent models of the convection-diffusion-reaction type. (C) 1998 Elsevie r Science B.V.