A stabilized nonconforming Nitsche's extended finite element method for Stokes interface problems

Xiaoxiao He; Fei Song; Weibing Deng

doi:10.3934/dcdsb.2021163

Article Contents

2022, Volume 27, Issue 5: 2849-2871. Doi: 10.3934/dcdsb.2021163

This issue Previous Article On q-deformed logistic maps Next Article On the mean field limit for Cucker-Smale models

A stabilized nonconforming Nitsche's extended finite element method for Stokes interface problems

1.
School of Science, Nanjing University, of Posts and Telecommunications, Nanjing, 210023, China
2.
School of Science, Nanjing Forestry University, Nanjing, 210037, China
3.
Department of Mathematics, Nanjing University, Nanjing, 210093, China

^* Corresponding author: Xiaoxiao He
^* Corresponding author: Xiaoxiao He

Received: December 31, 2020

Revised: March 31, 2021

Early access: June 2021

Published: May 2022

The first author is supported by NUPTSF (XK0070920088). The second author is partially supported by the Natural Science Foundation of Jiangsu Province (BK20190745) and the Natural Science Foundation of the Jiangsu Higher Institutions of China (18KJB110015) and the Youth Science and Technology Innovation Foundation of Nanjing Forestry University (CX2019026). The third author is partially supported by the the NSF of China grant 12090023, and by the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions

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Abstract

In this paper, a stabilized extended finite element method is proposed for Stokes interface problems on unfitted triangulation elements which do not require the interface align with the triangulation. The problem is written on mixed form using nonconforming $ P_1 $ velocity and elementwise $ P_0 $ pressure. Extra stabilization terms involving velocity and pressure are added in the discrete bilinear form. An inf-sup stability result is derived, which is uniform with respect to mesh size $ h $, the viscosity and the position of the interface. An optimal priori error estimates are obtained. Moreover, the errors in energy norm for velocity and in $ L^2 $ norm for pressure are uniform to the viscosity and the location of the interface. Results of numerical experiments are presented to support the theoretical analysis.

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Mathematics Subject Classification: Primary: 65N30; Secondary: 65N12, 65N15.

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Figure 1. A sample domain $ \Omega $

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Figure 2. Illustration of definitions of set $ G^{\Gamma}_{h} $, $ \Omega^{-}_{h,1} $, $ \Omega^{-}_{h,2} $, $ \Omega^{+}_{h,1} $ and $ \Omega^{+}_{h,2} $. Left figure: elements in $ G^{\Gamma}_{h} $(magenta area), $ \Omega^{-}_{h,1} $ and $ \Omega^{-}_{h,2} $ (cobalt blue area). Center figure: elements in $ \Omega^{+}_{h,1} $ (magenta area). Right figure: elements in $ \Omega^{+}_{h,2} $ (magenta area)

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Figure 3. Illustration of definitions of set $ \mathcal{F}^{nc}_{h,1} $, $ \mathcal{F}^{cut}_{h,1} $ and $ \mathcal{F}^{\Gamma}_{h,1} $. Left figure: edges in $ \mathcal{F}^{nc}_{h,1} $ (red lines). Center figure: edges in $ \mathcal{F}^{cut}_{h,1} $ (red lines). Right figure: edges in $ \mathcal{F}^{\Gamma}_{h,1} $ (red lines)

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Table 1. Errors for a continuous problem with $ \mu = 1 $

$ h $	$ e^1_{h,\mathbf{u}} $	rate	$ e^0_{h,\mathbf{u}} $	rate	$ e^0_{h,p} $	rate
$ 1/4 $	0.5040		0.2726		0.5597
$ 1/8 $	0.2816	0.8389	0.0920	1.5671	0.3237	0.7900
$ 1/16 $	0.1458	0.9497	0.0262	1.8121	0.1439	1.1696
$ 1/32 $	0.0737	0.9843	0.0066	1.9890	0.0615	1.2264
$ 1/64 $	0.0372	0.9864	0.0016	2.0444	0.0300	1.0356

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Table 2. Errors for an interface problem with $ \mu_1 = 1000 $ and $ \mu_2 = 1 $

$ h $	$ e^1_{h,\mathbf{u}} $	rate	$ e^0_{h,\mathbf{u}} $	rate	$ e^0_{h,p} $	rate
$ 1/4 $	0.5115		0.2754		0.5438
$ 1/8 $	0.2850	0.8438	0.0913	1.5928	0.2976	0.8697
$ 1/16 $	0.1463	0.9620	0.0253	1.8515	0.1503	0.9855
$ 1/32 $	0.0738	0.9872	0.0063	2.0057	0.0641	1.2294
$ 1/64 $	0.0373	0.9844	0.0016	1.9773	0.0302	1.0858

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Table 3. Errors for an interface problem with $ (\mu_1,\mu_2) = (10,1),(10^2,1),\cdots, (10^5,1) $ and fixed mesh $ h = 1/32 $

$ \mu_1 $	$ \mu_2 $	$ e^1_{h,\mathbf{u}} $	$ e^0_{h,\mathbf{u}} $	$ e^0_{h,p} $
$ 1E+01 $	$ 1 $	0.0738	0.0063	0.0598
$ 1E+02 $	$ 1 $	0.0737	0.0066	0.0612
$ 1E+03 $	$ 1 $	0.0737	0.0066	0.0615
$ 1E+04 $	$ 1 $	0.0737	0.0066	0.0615
$ 1E+05 $	$ 1 $	0.0737	0.0066	0.0615

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