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Optimal analysis of finite element methods for the stochastic Stokes equations
Authors
Buyang Li
Shu Ma
Weiwei Sun
Publication date
12 December 2023
Publisher
View
on
arXiv
Abstract
Numerical analysis for the stochastic Stokes equations is still challenging even though it has been well done for the corresponding deterministic equations. In particular, the pre-existing error estimates of finite element methods for the stochastic Stokes equations { in the
L
∞
(
0
,
T
;
L
2
(
Ω
;
L
2
)
)
L^\infty(0, T; L^2(\Omega; L^2))
L
∞
(
0
,
T
;
L
2
(
Ω
;
L
2
))
norm} all suffer from the order reduction with respect to the spatial discretizations. The best convergence result obtained for these fully discrete schemes is only half-order in time and first-order in space, which is not optimal in space in the traditional sense. The objective of this article is to establish strong convergence of
O
(
Ï„
1
/
2
+
h
2
)
O(\tau^{1/2}+ h^2)
O
(
Ï„
1/2
+
h
2
)
in the
L
∞
(
0
,
T
;
L
2
(
Ω
;
L
2
)
)
L^\infty(0, T; L^2(\Omega; L^2))
L
∞
(
0
,
T
;
L
2
(
Ω
;
L
2
))
norm for approximating the velocity, and strong convergence of
O
(
Ï„
1
/
2
+
h
)
O(\tau^{1/2}+ h)
O
(
Ï„
1/2
+
h
)
in the
L
∞
(
0
,
T
;
L
2
(
Ω
;
L
2
)
)
L^{\infty}(0, T;L^2(\Omega;L^2))
L
∞
(
0
,
T
;
L
2
(
Ω
;
L
2
))
norm for approximating the time integral of pressure, where
Ï„
\tau
Ï„
and
h
h
h
denote the temporal step size and spatial mesh size, respectively. The error estimates are of optimal order for the spatial discretization considered in this article (with MINI element), and consistent with the numerical experiments. The analysis is based on the fully discrete Stokes semigroup technique and the corresponding new estimates
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oai:arXiv.org:2206.13879
Last time updated on 26/08/2022