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

2015/12/21 by Mira Schedensack, Schedensack, Mira · 1 citation
Engineering · Mathematics · #31A30 #35J30 #65N12 #65N30 #74K20 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.1512.06513

openalex publication_date 2015/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper introduces new mixed formulations and discretizations for\nmth-Laplace equations of the form (-1)^m\Δ^m u=f for arbitrary\nm=1,2,3,\… based on novel Helmholtz-type decompositions for tensor-valued\nfunctions. The new discretizations allow for ansatz spaces of arbitrary\npolynomial degree and the lowest-order choice coincides with the non-conforming\nFEMs of Crouzeix and Raviart for m=1 and of Morley for m=2. Since the\nderivatives are directly approximated, the lowest-order discretizations consist\nof piecewise affine and piecewise constant functions for any m=1,2,\…\nMoreover, a uniform implementation for arbitrary m is possible. Besides the a\npriori and a posteriori analysis, this paper proves optimal convergence rates\nfor adaptive algorithms for the new discretizations.\n

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