A new discretization for mth-Laplace equations with arbitrary polynomial degrees
A new discretization for mth-Laplace equations with arbitrary polynomial degrees
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07.2016Publisher
Institut für Numerische Simulation (INS)Publisher Location
BonnMetadata
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Citable Link (Handle URI) 
https://hdl.handle.net/20.500.11811/11906
Abstract
This paper introduces new mixed formulations and discretizations for mth-Laplace equations of the form (−1)m∆mu = f for arbitrary m = 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 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.
Note
Revised version of December 2015Subjects
mth-Laplace equation, polyharmonic equation, non-conforming FEM, mixed FEM, adaptive FEM, optimality
Classification (DDC)
510 Mathematik 518 Numerische AnalysisFirst Publication
https://ins.uni-bonn.de/publication/preprintsRelated Publications
https://doi.org/10.1137/15M1013651Part of Series
INS Preprints ; 1528Schedensack, Mira: A new discretization for mth-Laplace equations with arbitrary polynomial degrees. In: INS Preprints, 1528.
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11906
Online-Ausgabe in bonndoc: https://hdl.handle.net/20.500.11811/11906
@unpublished{handle:20.500.11811/11906,
author = {{Mira Schedensack}},
title = {A new discretization for mth-Laplace equations with arbitrary polynomial degrees},
publisher = {Institut für Numerische Simulation (INS)},
year = 2016,
month = jul,
INS Preprints},
volume = 1528,
note = {This paper introduces new mixed formulations and discretizations for mth-Laplace equations of the form (−1)m∆mu = f for arbitrary m = 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 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.},
url = {https://hdl.handle.net/20.500.11811/11906}
}
author = {{Mira Schedensack}},
title = {A new discretization for mth-Laplace equations with arbitrary polynomial degrees},
publisher = {Institut für Numerische Simulation (INS)},
year = 2016,
month = jul,
INS Preprints},
volume = 1528,
note = {This paper introduces new mixed formulations and discretizations for mth-Laplace equations of the form (−1)m∆mu = f for arbitrary m = 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 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.},
url = {https://hdl.handle.net/20.500.11811/11906}
}
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