不可压缩Stokes方程解耦方法中的局部可计算$P^3$-压力_A locally calculable $P^3$-press
版权申诉
113 浏览量
2022-01-20
13:30:49
上传
评论
收藏 1.55MB PDF 举报
A locally calculable P
3
-pressure in a decoupled method for
incompressible Stokes equations
Chunjae Park
∗
Friday 19
th
November, 2021
Abstract
This paper will suggest a new finite element method to find a P
4
-velocity and a P
3
-pressure
solving incompressible Stokes equations at low cost. The method solves first the decoupled equation
for a P
4
-velocity. Then, using the calculated velocity, a locally calculable P
3
-pressure will be defined
component-wisely. The resulting P
3
-pressure is analyzed to have the optimal order of convergence.
Since the pressure is calculated by local computation only, the chief time cost of the new method is
on solving the decoupled equation for the P
4
-velocity. Besides, the method overcomes the problem
of singular vertices or corners.
1 Introduction
High order finite element methods for incompressible Stokes equations have been developed well in
2 dimensional domain and analyzed along with the inf-sup condition [1, 2, 6, 8, 11]. They, however,
endure their large degrees of freedom and have to avoid singular vertices or corners.
In the Scott-Vogelius finite element method, the inf-sup condition fails if the mesh has an exact
singular vertex. Even on nearly singular vertices, the pressure solution is easy to be spoiled. Recently,
to fix the problem, we have found a cause of singular vertex and devised a new error analysis based on
a so called sting function. As a result, the ruined pressure can be restored by simple post-process [9].
In this paper, employing the previous new error analysis, we will suggest a new finite element method
to find a P
4
-velocity and a P
3
-pressure solving incompressible Stokes equations at low cost.
The method will solve first the decoupled equation for a divergence-free P
4
-velocity which is almost
same as the one from the Falk-Neilan finite element method except corners [6]. Then, utilizing the
calculated velocity and orthogonal decomposition of P
3
, the 5 locally calculable components of a P
3
-
pressure will be defined by exploring locally calculable components in the Falk-Neilan and Scott-Vogelius
finite element spaces [6, 8, 11]. The resulting P
3
-pressure is analyzed to have the optimal order of
convergence.
Since the P
3
-pressure is calculated by local computation only, the chief time cost of the new method
is on solving the decoupled equation for the P
4
-velocity. If the pressure has a region of interest in Ω,
the regional computation is enough for it. Besides, the method overcomes the problem arising from the
singular vertices or corners by using the jump of the a priori calculated pressure components.
In the overall paper, the characteristics of sting functions depicted in Figure 1-(a) play key roles as
in the previous work in [9]. Since the sting function exists in P
k
for every integer k ≥ 0, the results for
P
4
− P
3
in this paper are easily extended for P
k+1
− P
k
, k ≥ 4 [10].
The paper is organized as follows. In the next section, the detail on finding a P
4
-velocity will be
offered. We will introduce an orthogonal decomposition of the space of P
3
-pressures, based on the
orthogonality of sting and non-sting functions in Section 3. Then, the most sections will be devoted
to defining the non-sting component for each triangle in Section 4 and the sting component for each
∗
1
arXiv:2107.08472v3 [math.NA] 21 Sep 2021
剩余21页未读，继续阅读
评论0
最新资源