hp-FEM for the $$\alpha $$-Mosolov problem: a priori and a posteriori error estimates

Abstract

An hp-finite element discretization for the $$\alpha $$ α -Mosolov problem, a scalar variant of the Bingham flow problem but with the $$\alpha $$ α -Laplacian operator, is being analyzed. Its weak formulation is either a variational inequality of second kind or equivalently a non-smooth but convex minimization problem. For any $$\alpha \in (1,\infty )$$ α ∈ ( 1 , ∞ ) we prove convergence, including guaranteed convergence rates in the mesh size h and polynomial degree p of the FE-solution of the corresponding discrete variational inequality. Moreover, we derive two families of reliable a posteriori error estimators which are applicable to any “approximation” of the exact solution and not only to the FE-solution and can therefore be coupled with an iterative solver. We prove that any quasi-minimizer of those families of a posteriori error estimators satisfies an efficiency estimate. All our results contain known results for the Mosolov problem by setting $$\alpha =2$$ α = 2 . Numerical results underline our theoretical findings.