Finite element approximation of degenerate quasilinear elliptic and parabolic problems

In this paper we present an overview of our recent work in this area. Energy type error bounds are derived for the finite element approximation of degenerate quasilinear elliptic equations of second order, which include the p-Laplacian. A key step in the analysis is to prove abstract error bounds initially in a quasi-norm, which naturally arises in degenerate problems of this type. These error bounds improve on existing results in the literature and are often optimal for sufficiently regular solutions, which we show to be achievable for a subclass of data. We extend these results to the corresponding parabolic problems and to the corresponding obstacle problems (variational inequalities). Finally, we extend these results to a non-Newtonian flow problem.