Numerical construction of optimal adaptive grids in two spatial dimensions

This article concerns a new procedure to generate a solution-adaptive grid for convection dominated problems in two spatial dimensions based on finite element approximations. The procedure extends a one-dimensional equidistribution principle which minimizes the interpolation error in appropriate norms. The idea in extending such a technique to two spatial dimensions is to select two directions which can reflect the physics of the problems, and then the one-dimensional equidistribution principle is applied to the chosen directions. The final grids generated are connected through a sweep-line based unstructured grid technique. Model problems considered are the two-dimensional convection-diffusion problems where boundary and interior layers occur. Numerical results of model problems illustrating the efficiency of the proposed scheme are presented. Comparisons of the solutions with an existing scheme will also be provided.