A new discretization for <em>m</em>th-Laplace equations with arbitrary polynomial degrees

Abstract

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