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

TL;DR

This paper develops new mixed discretizations for mth-Laplace equations using Helmholtz-type decompositions, enabling arbitrary polynomial degrees and uniform implementation, with proven optimal convergence rates.

Contribution

Introduces novel mixed formulations and discretizations for mth-Laplace equations based on Helmholtz-type decompositions, allowing arbitrary polynomial degrees and uniform implementation.

Findings

Discretizations match non-conforming FEMs for lowest order cases.

Derivatives are directly approximated with piecewise affine and constant functions.

Optimal convergence rates are proven for adaptive algorithms.

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 non-conforming 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.