The asymptotically optimal meshsize function for bi-p degree interpolation over rectangular elements

A method is presented to recover near optimal interpolation on finite element meshes based on information in the approximation error on an initial mesh. Only a certain class of admissable meshes with rectangular elements in the computational domains are allowed. The method attempts to reach the optimal mesh in one step from the initial mesh, and is based on the notion of meshsize function components or mesh density functions. Asymptotical results showing the optimality of the recovered meshes are given, and extensive computational verification of the method in the special case of Lagrange polynomial interpolation is provided.