A new discretization for mth-Laplace equations with arbitrary polynomial degrees

Schedensack M.

Abstract

This paper introduces new mixed formulations and discretizations for mth-Laplace equations of the form (-1)mΔmu = f for arbitrary m = 1,2,3,... based on novel Helmholtztype decompositions for tensor-valued functions. The new discretizations allow for ansatz spaces of arbitrary polynomial degree and the lowest-order choice coincides with the nonconforming FEMs of Crouzeix and Raviart for m = 1 and of Morley for m = 2. Since the derivatives are directly approximated, the lowest-order discretizations consist of piecewise affine and piecewise constant functions for any m = 1,2,.... Moreover, a uniform implementation for arbitrary m is possible. Besides the a priori and a posteriori analysis, this paper proves optimal convergence rates for adaptive algorithms for the new discretizations.