Anisotropic hp-mesh optimization technique based on the continuous mesh and error models

We develop a new mesh adaptive technique for the numerical solution of partial differential equations (PDEs) using the hp-version of the finite element method (hp-FEM). The technique uses a combination of approximation and interpolation error estimates to generate anisotropic triangular elements as well as appropriate polynomial approximation degrees. We present a hp-version of the continuous mesh model as well as the continuous error model which are used for the formulation of a mesh optimization problem. Solving the optimization problem leads to hp-mesh with the smallest number of degrees of freedom, under the constraint that the approximate solution has an error estimate below a given tolerance. Further, we propose an iterative algorithm to find a suitable anisotropic hp-mesh in the sense of the mesh optimization problem. Several numerical examples demonstrating the efficiency and applicability of the new method are presented.