Abstract
In this paper, we present a high-order weak Galerkin finite element method (WG-FEM) for solving the stationary Stokes interface problems with discontinuous velocity and pressure in ℝd, d = 2, 3. This WG method is equipped with stable finite elements consisting of usual polynomials of degree k ⩾ 1 for the velocity and polynomials of degree k − 1 for the pressure, both are discontinuous. Optimal convergence rates of order k + 1 for the velocity and order k for the pressure are established in L2-norm on hybrid meshes. Numerical experiments verify the expected order of accuracy for both two-dimensional and three-dimensional examples. Moreover, numerically it is shown that the proposed WG algorithm is able to accommodate geometrically complicated and very irregular interfaces having sharp edges, cusps, and tips.
Acknowledgement
The authors are grateful to the anonymous referee for valuable comments and suggestions which greatly improved the presentation of this paper.
References
[1] R. A. Adams and J. Fournier, Sobolev Spaces, Vol. 41, Academic Press, New York, 1975.Search in Google Scholar
[2] S. Adjerid, N. Chaabane, and T. Lin, An immersed discontinuous finite element method for Stokes interface problems, Computer Methods in Applied Mechanics and Engineering 293 (2015), 170–190.10.1016/j.cma.2015.04.006Search in Google Scholar
[3] R. F. Ausas, F. S. Sousa, and G. C. Buscaglia, An improved finite element space for discontinuous pressures, Computer Methods in Applied Mechanics and Engineering 199 (2010), No. 17-20, 1019–1031.10.1016/j.cma.2009.11.011Search in Google Scholar
[4] J. W. Barrett and C. M. Elliott, Fitted and unfitted finite-element methods for elliptic equations with smooth interfaces, IMA Journal of Numerical Analysis 7 (1987), No. 3, 283–300.10.1093/imanum/7.3.283Search in Google Scholar
[5] F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier–Stokes Equations and Related Models, Vol. 183, Springer Science & Business Media, 2012.10.1007/978-1-4614-5975-0Search in Google Scholar
[6] E. Burman, G. Delay, and A. Ern, An unfitted hybrid high-order method for the Stokes interface problem, IMA Journal of Numerical Analysis 41 (2021), No. 4, 2362–2387.10.1093/imanum/draa059Search in Google Scholar
[7] E. Burman and A. Ern, An unfitted hybrid high-order method for elliptic interface problems, SIAM Journal on Numerical Analysis 56 (2018), No. 3, 1525–1546.10.1137/17M1154266Search in Google Scholar
[8] W. Cao, C. Wang, and J. Wang, A new primal–dual weak Galerkin method for elliptic interface problems with low regularity assumptions, Journal of Computational Physics 470 (2022), 111538.10.1016/j.jcp.2022.111538Search in Google Scholar
[9] Y. Cao, M. Gunzburger, X. He, and X. Wang, Robin–Robin domain decomposition methods for the steady-state Stokes–Darcy system with the Beavers–Joseph interface condition, Numerische Mathematik 117 (2011), No. 4, 601–629.10.1007/s00211-011-0361-8Search in Google Scholar
[10] Y. Cao, M. Gunzburger, X. He, and X. Wang, Parallel, non-iterative, multi-physics domain decomposition methods for time-dependent Stokes–Darcy systems, Mathematics of Computation 83 (2014), No. 288, 1617–1644.10.1090/S0025-5718-2014-02779-8Search in Google Scholar
[11] C. Carstensen, Q. Zhai, and R. Zhang, A skeletal finite element method can compute lower eigenvalue bounds, SIAM Journal on Numerical Analysis 58 (2020), No. 1, 109–124.10.1137/18M1212276Search in Google Scholar
[12] Y.-C. Chang, T. Y. Hou, B. Merriman, and S. Osher, A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, Journal of Computational Physics 124 (1996), No. 2, 449–464.10.1006/jcph.1996.0072Search in Google Scholar
[13] L. Chen, H. Wei, and M. Wen, An interface-fitted mesh generator and virtual element methods for elliptic interface problems, J. Comput. Phys. 334 (2017), 327–348.10.1016/j.jcp.2017.01.004Search in Google Scholar
[14] W. Chen, F. Wang, and Y. Wang, Weak Galerkin method for the coupled Darcy–Stokes flow, IMA Journal of Numerical Analysis 36 (2016), No. 2, 897–921.10.1093/imanum/drv012Search in Google Scholar
[15] Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numerische Mathematik 79 (1998), No. 2, 175–202.Search in Google Scholar
[16] Z. Chen and J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 (1998), No. 2, 175–202.10.1007/s002110050336Search in Google Scholar
[17] J. M. Connors, J. S. Howell, and W. J. Layton, Decoupled time stepping methods for fluid–fluid interaction, SIAM Journal on Numerical Analysis 50 (2012), No. 3, 1297–1319.10.1137/090773362Search in Google Scholar
[18] J. M. Connors and B. Ganis, Stability of algorithms for a two domain natural convection problem and observed model uncertainty, Computational Geosciences 15 (2011), No. 3, 509–527.10.1007/s10596-010-9219-xSearch in Google Scholar
[19] B. Deka, Finite element methods with numerical quadrature for elliptic problems with smooth interfaces, J. Comput. Appl. Math. 234 (2010), No. 2, 605–612.10.1016/j.cam.2009.12.052Search in Google Scholar
[20] B. Deka, A weak Galerkin finite element method for elliptic interface problems with polynomial reduction, Numer. Math. Theory Methods Appl. 11 (2018), No. 3, 655–672.10.4208/nmtma.2017-OA-0078Search in Google Scholar
[21] S. Groß and A. Reusken, An extended pressure finite element space for two-phase incompressible flows with surface tension, Journal of Computational Physics 224 (2007), No. 1, 40–58.10.1016/j.jcp.2006.12.021Search in Google Scholar
[22] S. Gross and A. Reusken, Finite element discretization error analysis of a surface tension force in two-phase incompressible flows, SIAM journal on numerical analysis 45 (2007), No. 4, 1679–1700.10.1137/060667530Search in Google Scholar
[23] P. Hansbo, M. G. Larson, and S. Zahedi, A cut finite element method for a Stokes interface problem, Applied Numerical Mathematics 85 (2014), 90–114.10.1016/j.apnum.2014.06.009Search in Google Scholar
[24] X. He, J. Li, Y. Lin, and J. Ming, A domain decomposition method for the steady-state Navier–Stokes–Darcy model with Beavers–Joseph interface condition, SIAM Journal on Scientific Computing 37 (2015), No. 5, S264–S290.10.1137/140965776Search in Google Scholar
[25] G. Hou, J. Wang, and A. Layton, Numerical methods for fluid–structure interaction — A review, Communications in Computational Physics 12 (2012), No. 2, 337–377.10.4208/cicp.291210.290411sSearch in Google Scholar
[26] L. N. T. Huynh, N. C. Nguyen, J. Peraire, and B. C. Khoo, A high-order hybridizable discontinuous Galerkin method for elliptic interface problems, Int. J. Numer. Methods Engrg. 93 (2013), No. 2, 183–200.10.1002/nme.4382Search in Google Scholar
[27] D. Jones and X. Zhang, A class of nonconforming immersed finite element methods for Stokes interface problems, Journal of Computational and Applied Mathematics 392 (2021), 113493.10.1016/j.cam.2021.113493Search in Google Scholar
[28] A. Khan, C. S. Upadhyay, and M. Gerritsma, Spectral element method for parabolic interface problems, Computer Methods in Applied Mechanics and Engineering 337 (2018), 66–94.10.1016/j.cma.2018.03.011Search in Google Scholar
[29] G. Laymuns and M. A. Sánchez, Corrected finite element methods on unfitted meshes for Stokes moving interface problem, Computers & Mathematics with Applications 108 (2022), 159–174.10.1016/j.camwa.2021.12.018Search in Google Scholar
[30] C. L. Brossier, V. Ducrocq, and H. Giordani, Effects of the air–sea coupling time frequency on the ocean response during Mediterranean intense events, Ocean Dynamics 59 (2009), 539–549.10.1007/s10236-009-0198-1Search in Google Scholar
[31] L. Lee and R. J. LeVeque, An immersed interface method for incompressible Navier–Stokes equations, SIAM Journal on Scientific Computing 25 (2003), No. 3, 832–856.10.1137/S1064827502414060Search in Google Scholar
[32] R. J. LeVeque and Z. Li, Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM Journal on Scientific Computing 18 (1997), No. 3, 709–735.10.1137/S1064827595282532Search in Google Scholar
[33] J. Li, J. M. Melenk, B. Wohlmuth, and J. Zou, Optimal a priori estimates for higher order finite elements for elliptic interface problems, Applied Numerical Mathematics 60 (2010), No. 1-2, 19–37.10.1016/j.apnum.2009.08.005Search in Google Scholar
[34] Z. Li and M.-C. Lai, The immersed interface method for the Navier–Stokes equations with singular forces, Journal of Computational Physics 171 (2001), No. 2, 822–842.10.1006/jcph.2001.6813Search in Google Scholar
[35] L. Mu, Pressure robust weak Galerkin finite element methods for Stokes problems, SIAM Journal on Scientific Computing 42 (2020), No. 3, B608–B629.10.1137/19M1266320Search in Google Scholar
[36] L. Mu, J. Wang, G. Wei, X. Ye, and S. Zhao, Weak Galerkin methods for second order elliptic interface problems, Journal of Computational Physics 250 (2013), 106–125.10.1016/j.jcp.2013.04.042Search in Google Scholar PubMed PubMed Central
[37] L. Mu, J. Wang, X. Ye, and S. Zhao, A new weak Galerkin finite element method for elliptic interface problems, J. Comput. Phys. 325 (2016), 157–173.Search in Google Scholar
[38] L. Mu, J. Wang, X. Ye, and S. Zhao, A new weak Galerkin finite element method for elliptic interface problems, Journal of Computational Physics 325 (2016), 157–173.10.1016/j.jcp.2016.08.024Search in Google Scholar
[39] J. Nelson, R. He, J. C. Warner, and J. Bane, Air–sea interactions during strong winter extratropical storms, Ocean Dynamics 64 (2014), 1233–1246.10.1007/s10236-014-0745-2Search in Google Scholar
[40] K. Ohmori and N. Saito, On the convergence of finite element solutions to the interface problem for the Stokes system, Journal of Computational and Applied Mathematics 198 (2007), No. 1, 116–128.10.1016/j.cam.2005.11.018Search in Google Scholar
[41] M. A. Olshanskii, J. Peters, and A. Reusken, Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface equations, Numerische Mathematik 105 (2006), 159–191.10.1007/s00211-006-0031-4Search in Google Scholar
[42] M. A. Olshanskii and A. Reusken, Analysis of a Stokes interface problem, Numerische Mathematik 103 (2006), No. 1, 129–149.10.1007/s00211-005-0646-xSearch in Google Scholar
[43] M. Plum and C. Wieners, Optimal a priori estimates for interface problems, Numerische Mathematik 95 (2003), No. 4, 735–759.10.1007/s002110200395Search in Google Scholar
[44] M. Shao, L. Song, and P.-W. Li, A generalized finite difference method for solving Stokes interface problems, Engineering Analysis with Boundary Elements 132 (2021), 50–64.10.1016/j.enganabound.2021.07.002Search in Google Scholar
[45] Y. Shibata and S. Shimizu, On a resolvent estimate of the interface problem for the Stokes system in a bounded domain, Journal of Differential Equations 191 (2003), No. 2, 408–444.10.1016/S0022-0396(03)00023-8Search in Google Scholar
[46] P. Sun, Fictitious domain finite element method for Stokes/elliptic interface problems with jump coefficients, Journal of Computational and Applied Mathematics 356 (2019), 81–97.10.1016/j.cam.2019.01.030Search in Google Scholar
[47] J. Tushar, A. Kumar, and S. Kumar, Virtual element methods for general linear elliptic interface problems on polygonal meshes with small edges, Computers & Mathematics with Applications 122 (2022), 61–75.10.1016/j.camwa.2022.07.016Search in Google Scholar
[48] I. Voulis and A. Reusken, A time dependent Stokes interface problem: well-posedness and space-time finite element discretization, ESAIM: Mathematical Modelling and Numerical Analysis 52 (2018), No. 6, 2187–2213.10.1051/m2an/2018053Search in Google Scholar
[49] J. Wang and X. Ye, A weak Galerkin mixed finite element method for second order elliptic problems, Mathematics of Computation 83 (2014), No. 289, 2101–2126.10.1090/S0025-5718-2014-02852-4Search in Google Scholar
[50] J. Wang and X. Ye, A weak Galerkin finite element method for the Stokes equations, Advances in Computational Mathematics 42 (2016), No. 1, 155–174.10.1007/s10444-015-9415-2Search in Google Scholar
[51] N. Wang and J. Chen, A nonconforming Nitsche’s extended finite element method for Stokes interface problems, Journal of Scientific Computing 81 (2019), 342–374.10.1007/s10915-019-01019-9Search in Google Scholar
[52] Q. Wang and J. Chen, A new unfitted stabilized Nitsche’s finite element method for Stokes interface problems, Computers & Mathematics with Applications 70 (2015), No. 5, 820–834.10.1016/j.camwa.2015.05.024Search in Google Scholar
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