Two-grid finite element method for the stabilization of mixed Stokes-Darcy model

Jiaping Yu; Haibiao Zheng; Feng Shi; Ren Zhao

doi:10.3934/dcdsb.2018109

Article Contents

2019, Volume 24, Issue 1: 387-402. Doi: 10.3934/dcdsb.2018109

This issue Previous Article Long-time dynamics for a non-autonomous Navier-Stokes-Voigt equation in Lipschitz domains Next Article Lower and upper bounds of Laplacian eigenvalue problem by weak Galerkin method on triangular meshes

Two-grid finite element method for the stabilization of mixed Stokes-Darcy model

1.
College of Science, Donghua University, Shanghai 201620, China
2.
Department of Mathematics, East China Normal University, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, Shanghai 200241, China
3.
College of Science, Harbin Institute of Technology Shenzhen Graduate School, Shenzhen 518055, China
4.
Department of Mathematics, University of Houston, Houston, TX 77024, USA
5.
Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

* Corresponding author: Feng Shi (shi.feng@hit.edu.cn)
* Corresponding author: Feng Shi (shi.feng@hit.edu.cn)

Received: March 31, 2017

Revised: July 31, 2017

Early access: March 2018

Published: January 2019

The first author is supported by NSFC (Grant Nos. 11501097 and 11471071). The second author is partially supported by NSFC (Grant Nos. 11201369 and 11771337). The third author is partially subsidized by Basic Research Program of Shenzhen (Grant No. JCYJ20150831112754988).

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Abstract

A two-grid discretization for the stabilized finite element method for mixed Stokes-Darcy problem is proposed and analyzed. The lowest equal-order velocity-pressure pairs are used due to their simplicity and attractive computational properties, such as much simpler data structures and less computer memory for meshes and algebraic system, easier interpolations, and convenient usages of many existing preconditioners and fast solvers in simulations, which make these pairs a much popular choice in engineering practice; see e.g., [4,27]. The decoupling methods are adopted for solving coupled systems based on the significant features that decoupling methods can allow us to solve the submodel problems independently by using most appropriate numerical techniques and preconditioners, and also to reduce substantial coding tasks. The main idea in this paper is that, on the coarse grid, we solve a stabilized finite element scheme for coupled Stokes-Darcy problem; then on the fine grid, we apply the coarse grid approximation to the interface conditions, and solve two independent subproblems: one is the stabilized finite element method for Stokes subproblem, and another one is the Darcy subproblem. Optimal error estimates are derived, and several numerical experiments are carried out to demonstrate the accuracy and efficiency of the two-grid stabilized finite element algorithm.

Keywords:

Mathematics Subject Classification: Primary: 65N30, 65N15; Secondary: 76D07.

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Figure 1. The pressure line by TGM (Left), StbTGM (Middle) and TGM-$(P_1, P_1, P_1)$ (Right)

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Figure 2. Streamlines for the numerical velocity by TGM (Left), StbTGM (Middle) and TGM-$(P_1, P_1, P_1)$ (Right)

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Figure 3. The velocity streamlines of the backward facing step flow with two interface conditions: Case 1 (top), Case 2 (bottom)

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Table 1. The convergence performance and CPU time by SFEM

h	$\\|\varphi-\varphi^h\\|_{1, \Omega_p}$	Rate	$\\|u-u^h\\|_{1, \Omega_f}$	Rate	$\\|p-p^h\\|_{0, \Omega_f}$	Rate	CPU
$\frac14$	3.666e-1	-	2.051e-1	-	9.850e-1	-	0.101
$\frac{1}{16}$	9.850e-2	0.948	5.103e-2	1.004	9.534e-2	1.684	3.547
$\frac{1}{64}$	2.476e-2	0.996	1.280e-2	0.998	3.858e-2	0.653	214.951

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Table 2. The convergence performance and CPU time by TGM

H	h	$\\|\varphi-\varphi^h\\|_{1, \Omega_p}$	Rate	$\\|u-u^h\\|_{1, \Omega_f}$	Rate	$\\|p-p^h\\|_{0, \Omega_f}$	Rate	CPU
$\frac14$	$\frac{1}{16}$	9.577e-2	-	5.109e-2	-	1.741e-1	-	0.181
$\frac{1}{8}$	$\frac{1}{64}$	2.405e-2	0.997	1.275e-2	1.001	3.799e-2	1.098	2.127
$\frac{1}{16}$	$\frac{1}{256}$	6.014e-3	1.000	3.187e-3	1.000	9.134e-3	1.028	39.827

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Table 3. The convergence performance and CPU time by StbTGM

H	h	$\\|\varphi-\varphi^h\\|_{1, \Omega_p}$	Rate	$\\|u-u^h\\|_{1, \Omega_f}$	Rate	$\\|p-p^h\\|_{0, \Omega_f}$	Rate	CPU
$\frac{1}{4}$	$\frac{1}{16}$	9.577e-2	-	5.494e-2	-	1.540e-1	-	0.117
$\frac{1}{8}$	$\frac{1}{64}$	2.404e-2	0.997	1.375e-2	1.000	3.682e-2	1.032	1.351
$\frac{1}{16}$	$\frac{1}{256}$	6.012e-3	1.000	3.437e-3	1.000	9.267e-3	0.995	26.805

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Table 4. The convergence performance by StbTGM with fixed $H = 1/8$

H	h	$\\|\varphi-\varphi^h\\|_{1, \Omega_p}$	Rate	$\\|u-u^h\\|_{1, \Omega_f}$	Rate	$\\|p-p^h\\|_{0, \Omega_f}$	Rate
$\frac18$	$\frac{1}{16}$	9.470e-2		5.489e-2		5.040e-2
$\frac18$	$\frac{1}{64}$	2.404e-2	0.989	1.375e-2	0.999	3.682e-2	0.226
$\frac18$	$\frac{1}{256}$	7.033e-3	0.887	3.525e-3	0.982	3.704e-2	-0.004

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Table 5. The convergence performance and CPU time by TGM-$(P_1, P_1, P_1)$

H	h	$\\|\varphi-\varphi^h\\|_{1, \Omega_p}$	Rate	$\\|u-u^h\\|_{1, \Omega_f}$	Rate	$\\|p-p^h\\|_{0, \Omega_f}$	Rate	CPU
$\frac{1}{4}$	$\frac{1}{16}$	9.553e-2	-	5.539e-2	-	4.594e+7	-	0.117
$\frac{1}{8}$	$\frac{1}{64}$	2.403e-2	0.996	1.386e-2	0.999	2.904e+6	-	1.351
$\frac{1}{16}$	$\frac{1}{256}$	null	-	null	-	null	-	-

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Table 6. The approximation errors by StbTGM for $k = 0.1$

H	h	$\\|\varphi-\varphi^h\\|_{1, \Omega_p}$	Rate	$\\|u-u^h\\|_{1, \Omega_f}$	Rate	$\\|p-p^h\\|_{0, \Omega_f}$	Rate
$\frac{1}{4}$	$\frac{1}{16}$	5.181e-2	-	4.057e-2	-	2.15254e-2	-
$\frac{1}{8}$	$\frac{1}{64}$	1.297e-2	0.999	1.006e-2	1.006	5.853e-3	0.939
$\frac{1}{16}$	$\frac{1}{256}$	3.243e-3	1.000	2.509e-3	1.001	1.536e-3	0.965

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Table 7. The approximation errors by StbTGM for $k = 0.01$

H	h	$\\|\varphi-\varphi^h\\|_{1, \Omega_p}$	Rate	$\\|u-u^h\\|_{1, \Omega_f}$	Rate	$\\|p-p^h\\|_{0, \Omega_f}$	Rate
$\frac{1}{4}$	$\frac{1}{16}$	5.397e-2	-	7.288e-2	-	3.03543e-2	-
$\frac{1}{8}$	$\frac{1}{64}$	1.350e-2	1.000	1.610e-2	1.089	8.070e-3	0.956
$\frac{1}{16}$	$\frac{1}{256}$	3.372e-3	1.001	3.785e-3	1.044	2.081e-3	0.978

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Table 8. The approximation errors by StbTGM for $k = 0.001$

H	h	$\\|\varphi-\varphi^h\\|_{1, \Omega_p}$	Rate	$\\|u-u^h\\|_{1, \Omega_f}$	Rate	$\\|p-p^h\\|_{0, \Omega_f}$	Rate
$\frac{1}{4}$	$\frac{1}{16}$	5.490e-2	-	4.467e-1	-	2.793e-2	-
$\frac{1}{8}$	$\frac{1}{64}$	1.375e-2	0.999	6.834e-2	1.3541	6.980e-3	1.000
$\frac{1}{16}$	$\frac{1}{256}$	3.423e-3	1.003	1.229e-2	1.238	1.755e-3	0.996

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