Anisotropic mesh refinement for finite element methods based on error reduction

In this paper, we propose an anisotropic adaptive refinement algorithm based on the finite element methods for the numerical solution of partial differential equations. In 2-D, for a given triangular grid and finite element approximating space V, we obtain information on location and direction of refinement by estimating the reduction of the error if a single degree of freedom is added to V. For our model problem the algorithm fits highly stretched triangles along an interior layer, reducing the number of degrees of freedom that a standard h-type isotropic refinement algorithm would use.