A New Discretization for $m$th-Laplace Equations with Arbitrary Polynomial Degrees

Abstract

This paper introduces new mixed formulations and discretizations for $m$th-Laplace equations of the form $(-1)^m\Delta^m u=f$ for arbitrary $m=1,2,3,\dots$ 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 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 $m=1,2,\dots.$ 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.