# American Institute of Mathematical Sciences

January  2019, 24(1): 387-402. doi: 10.3934/dcdsb.2018109

## 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)

Received  April 2017 Revised  August 2017 Published  March 2018

Fund Project: 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)

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.

Citation: Jiaping Yu, Haibiao Zheng, Feng Shi, Ren Zhao. Two-grid finite element method for the stabilization of mixed Stokes-Darcy model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 387-402. doi: 10.3934/dcdsb.2018109
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##### References:
The pressure line by TGM (Left), StbTGM (Middle) and TGM-$(P_1, P_1, P_1)$ (Right)
Streamlines for the numerical velocity by TGM (Left), StbTGM (Middle) and TGM-$(P_1, P_1, P_1)$ (Right)
The velocity streamlines of the backward facing step flow with two interface conditions: Case 1 (top), Case 2 (bottom)
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
 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
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
 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
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
 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
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
 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
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 - -
 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 - -
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
 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
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
 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
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
 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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