r-refinement for evolutionary PDEs with finite elements or finite differences Original Research Article

In this paper two different moving-mesh methods (r-refinement) are applied to evolutionary PDE models in one and two space dimensions. The first method (moving finite elements) is based on a minimization of the PDE residual that is obtained by approximating the solution with piecewise linear elements. The second method (moving finite differences) is based on an equidistribution principle with smoothing both in the spatial and the temporal direction. Theory predicts that the finite-element based moving-mesh method moves its grid points with the flow of the PDE, whereas the finite-difference based method moves its grid points with the steep parts of the PDE solution, respectively. Numerical experiments show some differences and similarities between the finite-element and finite-difference case when applied to 1D and 2D time-dependent models of the convection—diffusion—reaction type.