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research
W
s
,
p
W^{s,p}
W
s
,
p
-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray-Lions problems
Authors
Daniele Di Pietro
Jerome Droniou
Publication date
1 January 2017
Publisher
Doi
Cite
View
on
arXiv
Abstract
In this work we prove optimal
W
s
,
p
W^{s,p}
W
s
,
p
-approximation estimates (with
p
∈
[
1
,
+
∞
]
p\in[1,+\infty]
p
∈
[
1
,
+
∞
]
) for elliptic projectors on local polynomial spaces. The proof hinges on the classical Dupont--Scott approximation theory together with two novel abstract lemmas: An approximation result for bounded projectors, and an
L
p
L^p
L
p
-boundedness result for
L
2
L^2
L
2
-orthogonal projectors on polynomial subspaces. The
W
s
,
p
W^{s,p}
W
s
,
p
-approximation results have general applicability to (standard or polytopal) numerical methods based on local polynomial spaces. As an illustration, we use these
W
s
,
p
W^{s,p}
W
s
,
p
-estimates to derive novel error estimates for a Hybrid High-Order discretization of Leray--Lions elliptic problems whose weak formulation is classically set in
W
1
,
p
(
Ω
)
W^{1,p}(\Omega)
W
1
,
p
(
Ω
)
for some
p
∈
(
1
,
+
∞
)
p\in(1,+\infty)
p
∈
(
1
,
+
∞
)
. This kind of problems appears, e.g., in the modelling of glacier motion, of incompressible turbulent flows, and in airfoil design. Denoting by
h
h
h
the meshsize, we prove that the approximation error measured in a
W
1
,
p
W^{1,p}
W
1
,
p
-like discrete norm scales as
h
k
+
1
p
−
1
h^{\frac{k+1}{p-1}}
h
p
−
1
k
+
1
​
when
p
≥
2
p\ge 2
p
≥
2
and as
h
(
k
+
1
)
(
p
−
1
)
h^{(k+1)(p-1)}
h
(
k
+
1
)
(
p
−
1
)
when
p
<
2
p<2
p
<
2
.Comment: keywords:
W
s
,
p
W^{s,p}
W
s
,
p
-approximation properties of elliptic projector on polynomials, Hybrid High-Order methods, nonlinear elliptic equations,
p
p
p
-Laplacian, error estimate
Similar works
Full text
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HAL Descartes
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oai:HAL:hal-01326818v1
Last time updated on 14/04/2021
Monash University Research Portal
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Go to the repository landing page
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oai:monash.edu:publications/27...
Last time updated on 05/12/2019