A New Multilevel Homotopic Adaptive Finite Element Algorithm for Convection-Dominated Diffusion-Reaction Problems

Abstract

The multilevel homotopic adaptive finite element method (MHAFEM) was developed for solving convection-dominated diffusion-reaction problems on triangular meshes, where the mesh adaptation (refinement and mesh smoothing/moving) is based on an optimal interpolation error estimate in $L^p$ norm. In this paper, we follow the framework of the MHAFEM to give a new multilevel homotopic adaptive finite element algorithm. This new MHAFEM executes mesh refinement based on an a posteriori error estimator instead of the interpolation error estimate. We apply this algorithm to 1D and 2D example problems using 1D linear elements and 2D quadrilateral transition elements, respectively. Numerical results demonstrate the remarkable efficiency of the algorithm.