# American Institute of Mathematical Sciences

May  2022, 27(5): 2849-2871. doi: 10.3934/dcdsb.2021163

## 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

Received  January 2021 Revised  April 2021 Published  May 2022 Early access  June 2021

Fund Project: 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

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.

Citation: Xiaoxiao He, Fei Song, Weibing Deng. A stabilized nonconforming Nitsche's extended finite element method for Stokes interface problems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2849-2871. doi: 10.3934/dcdsb.2021163
##### References:
 [1] S. Adjerid, N. Chaabane and T. Lin, An immersed discontinuous finite element method for Stokes interface problems, Comput. Methods Appl. Mech. Engrg., 293 (2015), 170-190.  doi: 10.1016/j.cma.2015.04.006. [2] N. Barrau, R. Becker, E. Dubach and R. Luce, A robust variant of NXFEM for the interface problem, C. R. Math. Acad. Sci. Paris, 350 (2012), 789-792.  doi: 10.1016/j.crma.2012.09.018. [3] R. Becker, E. Burman and P. Hansbo, A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity, Comput. Methods Appl. Mech. Engrg., 198 (2009), 3352-3360.  doi: 10.1016/j.cma.2009.06.017. [4] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3$^rd$ edition, Springer-Verlag, 2008. doi: 10.1007/978-0-387-75934-0. [5] E. Burman, Ghost penalty, C. R. Math. Acad. Sci. Paris, 348 (2010), 1217-1220.  doi: 10.1016/j.crma.2010.10.006. [6] E. Burman, J. Guzmán, M. A. Sánchez and M. Sarkis, Robust flux error estimation of Nitsche's method for high contrast interface problems, IMA J. Numer. Anal., 38 (2018), 646-668.  doi: 10.1093/imanum/drx017. [7] E. Burman and P. Zunino, A domain decomposition method based on weighted interior penalties for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 44 (2006), 1612-1638.  doi: 10.1137/050634736. [8] Z. Cai, X. Ye and S. Zhang, Discontinuous Galerkin finite element methods for interface problems: a priori and a posteriori error estimations, SIAM J. Numer. Anal., 49 (2011), 1761-1787.  doi: 10.1137/100805133. [9] D. Capatina, S. Delage Sabtacrey, H. EI-Otmany and D. Graebling, Nonconforming finite element approximation of an elliptic interface problem with NXFEM, Thirteenth International Conference Zaragoza-Pau on Mathematics and its Applications (Monogr. Mat. García Galdeano), Prensas Univ. Zaragoza, Zaragoza, 40 2016, 43–52. [10] D. Capatina, H. EI-Otmany, D. Graebling and R. Luce, Extension of NXFEM to nonconforming finite elements, Math. Comput. Simulation, 137 (2017), 226-245.  doi: 10.1016/j.matcom.2016.12.009. [11] L. Cattaneo, L. Formaggia, G. F. Iori, A. Scotti and P. Zunino, Stabilized extended finite elements for the approximation of saddle point problems with unfitted interfaces, Calcolo, 52 (2015), 123-152.  doi: 10.1007/s10092-014-0109-9. [12] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. [13] H. EI-Otmany, Approximation by NXFEM Method of Interphase and Interface Problems in Fluid Mechanics, Ph.D thesis, November 2015. doi: 10.13140/RG.2.1.2949.6403. [14] A. Ern, A. F. Stephansen and P. Zunino, A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity, IMA J. Numer. Anal., 29 (2009), 235-256.  doi: 10.1093/imanum/drm050. [15] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5. [16] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5. [17] Y. Gong, B. Li and Z. Li, Immersed-interface finite-element method for elliptic interface problems with nonhomogeneous jump conditions, SIAM J. Numer. Anal., 46 (2007/08), 472-495.  doi: 10.1137/060666482. [18] J. Guzmán, M. A. Sánchez and M. Sarkis, A finite element method for high-contrast interface problems with error estimates independent of contrast, J. Sci. Comput., 73 (2017), 330-365.  doi: 10.1007/s10915-017-0415-x. [19] A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche's method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg., 191 (2002), 5537-5552.  doi: 10.1016/S0045-7825(02)00524-8. [20] P. Hansbo, M. G. Larson and S. Zahedi, A cut finite element method for a Stokes interface problem, Appl. Numer. Math., 85 (2014), 90-114.  doi: 10.1016/j.apnum.2014.06.009. [21] X. He, W. Deng and H. Wu, An interface penalty finite element method for elliptic interface problems on piecewise meshes, J. Comput. Appl. Math., 367 (2020), 112473, 20. doi: 10.1016/j.cam.2019.112473. [22] X. He, F. Song and W. Deng, A well-conditioned, nonconforming Nitsche's extended finite element method for elliptic interface problems, Numer. Math. Theory Methods Appl., 13 (2020), 99-130.  doi: 10.4208/nmtma.OA-2019-0053. [23] H. Huang and Z. Li, Convergence analysis of the immersed interface method, IMA J. Numer. Anal., 19 (1999), 583-608.  doi: 10.1093/imanum/19.4.583. [24] P. Huang, H. Wu and Y. Xiao, An unfitted interface penalty finite element method for elliptic interface problems, Comput. Methods Appl. Mech. Engrg., 323 (2017), 439-460.  doi: 10.1016/j.cma.2017.06.004. [25] M. Kirchhart, S. Gross and A. Reusken, Analysis of an XFEM discretization for Stokes interface problems, SIAM J. Sci. Comput., 38 (2016), 1019-1043.  doi: 10.1137/15M1011779. [26] D. Y. Kwak, K. T. Wee and K. S. Chang, An analysis of broken $P_1$-nonconforming finite element method for interface problems, SIAM J. Numer. Anal., 48 (2010), 2117-2134.  doi: 10.1137/080728056. [27] Z. Li, T. Lin and X. Wu, New Cartesian grid methods for interface problems using the finite element formulation, Numer. Math., 96 (2003), 61-98. doi: 10.1007/s00211-003-0473-x. [28] T. Lin, D. Sheen and X. Zhang, A nonconforming immersed finite element method for elliptic interface problems, J. Sci. Comput., 79 (2019), 442-463.  doi: 10.1007/s10915-018-0865-9. [29] T. Lin, Q. Yang and X. Zhang, Partially penalized immersed finite element methods for parabolic interface problems, Numer. Methods Partial Differential Equations, 31 (2015), 1925-1947.  doi: 10.1002/num.21973. [30] A. Massing, M. G. Larson, A. Logg and M. E. Rognes, A stabilized Nitsche fictitious domain method for the Stokes problem, J. Sci. Comput., 61 (2014), 604-628.  doi: 10.1007/s10915-014-9838-9. [31] A. Massing, B. Schott and W. A. Wall, A stabilized Nitsche cut finite element method for the Oseen problem, Comput. Methods Appl. Mech. Engrg., 328 (2018), 262-300.  doi: 10.1016/j.cma.2017.09.003. [32] R. Massjung, An unfitted discontinuous Galerkin method applied to elliptic interface problems, SIAM J. Numer. Anal., 50 (2012), 3134-3162.  doi: 10.1137/090763093. [33] M. Sarkis, Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements, Numer. Math., 77 (1997), 383-406.  doi: 10.1007/s002110050292. [34] E. Wadbro, S. Zahedi, G. Kreiss and M. Berggren, A uniformly well-conditioned, unfitted Nitsche method for interface problems, Bit Numerical Mathematics, 53 (2013), 791-820.  doi: 10.1007/s10543-012-0417-x. [35] N. Wang and J. Chen, A nonconforming Nitsche's extended finite element method for Stokes interface problems, J. Sci. Comput., 81 (2019), 342-374.  doi: 10.1007/s10915-019-01019-9. [36] Q. Wang and J. Chen, A new unfitted stabilized Nitsche's finite element method for Stokes interface problems, Comput. Math. Appl., 70 (2015), 820-834.  doi: 10.1016/j.camwa.2015.05.024. [37] H. Wu and Y. Xiao, An unfitted $hp$-interface penalty finite element method for elliptic interface problems, J. Comput. Math., 37 (2019), 316-339.  doi: 10.4208/jcm.1802-m2017-0219. [38] Y. Xiao, J. Xu and F. Wang, High-order extended finite element methods for solving interface problems, Comput. Methods Appl. Mech. Engrg., 364 (2020), 112964, 21 pp. doi: 10.1016/j.cma.2020.112964.

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##### References:
 [1] S. Adjerid, N. Chaabane and T. Lin, An immersed discontinuous finite element method for Stokes interface problems, Comput. Methods Appl. Mech. Engrg., 293 (2015), 170-190.  doi: 10.1016/j.cma.2015.04.006. [2] N. Barrau, R. Becker, E. Dubach and R. Luce, A robust variant of NXFEM for the interface problem, C. R. Math. Acad. Sci. Paris, 350 (2012), 789-792.  doi: 10.1016/j.crma.2012.09.018. [3] R. Becker, E. Burman and P. Hansbo, A Nitsche extended finite element method for incompressible elasticity with discontinuous modulus of elasticity, Comput. Methods Appl. Mech. Engrg., 198 (2009), 3352-3360.  doi: 10.1016/j.cma.2009.06.017. [4] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3$^rd$ edition, Springer-Verlag, 2008. doi: 10.1007/978-0-387-75934-0. [5] E. Burman, Ghost penalty, C. R. Math. Acad. Sci. Paris, 348 (2010), 1217-1220.  doi: 10.1016/j.crma.2010.10.006. [6] E. Burman, J. Guzmán, M. A. Sánchez and M. Sarkis, Robust flux error estimation of Nitsche's method for high contrast interface problems, IMA J. Numer. Anal., 38 (2018), 646-668.  doi: 10.1093/imanum/drx017. [7] E. Burman and P. Zunino, A domain decomposition method based on weighted interior penalties for advection-diffusion-reaction problems, SIAM J. Numer. Anal., 44 (2006), 1612-1638.  doi: 10.1137/050634736. [8] Z. Cai, X. Ye and S. Zhang, Discontinuous Galerkin finite element methods for interface problems: a priori and a posteriori error estimations, SIAM J. Numer. Anal., 49 (2011), 1761-1787.  doi: 10.1137/100805133. [9] D. Capatina, S. Delage Sabtacrey, H. EI-Otmany and D. Graebling, Nonconforming finite element approximation of an elliptic interface problem with NXFEM, Thirteenth International Conference Zaragoza-Pau on Mathematics and its Applications (Monogr. Mat. García Galdeano), Prensas Univ. Zaragoza, Zaragoza, 40 2016, 43–52. [10] D. Capatina, H. EI-Otmany, D. Graebling and R. Luce, Extension of NXFEM to nonconforming finite elements, Math. Comput. Simulation, 137 (2017), 226-245.  doi: 10.1016/j.matcom.2016.12.009. [11] L. Cattaneo, L. Formaggia, G. F. Iori, A. Scotti and P. Zunino, Stabilized extended finite elements for the approximation of saddle point problems with unfitted interfaces, Calcolo, 52 (2015), 123-152.  doi: 10.1007/s10092-014-0109-9. [12] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. [13] H. EI-Otmany, Approximation by NXFEM Method of Interphase and Interface Problems in Fluid Mechanics, Ph.D thesis, November 2015. doi: 10.13140/RG.2.1.2949.6403. [14] A. Ern, A. F. Stephansen and P. Zunino, A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity, IMA J. Numer. Anal., 29 (2009), 235-256.  doi: 10.1093/imanum/drm050. [15] A. Ern and J.-L. Guermond, Theory and Practice of Finite Elements, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5. [16] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5. [17] Y. Gong, B. Li and Z. Li, Immersed-interface finite-element method for elliptic interface problems with nonhomogeneous jump conditions, SIAM J. Numer. Anal., 46 (2007/08), 472-495.  doi: 10.1137/060666482. [18] J. Guzmán, M. A. Sánchez and M. Sarkis, A finite element method for high-contrast interface problems with error estimates independent of contrast, J. Sci. Comput., 73 (2017), 330-365.  doi: 10.1007/s10915-017-0415-x. [19] A. Hansbo and P. Hansbo, An unfitted finite element method, based on Nitsche's method, for elliptic interface problems, Comput. Methods Appl. Mech. Engrg., 191 (2002), 5537-5552.  doi: 10.1016/S0045-7825(02)00524-8. [20] P. Hansbo, M. G. Larson and S. Zahedi, A cut finite element method for a Stokes interface problem, Appl. Numer. Math., 85 (2014), 90-114.  doi: 10.1016/j.apnum.2014.06.009. [21] X. He, W. Deng and H. Wu, An interface penalty finite element method for elliptic interface problems on piecewise meshes, J. Comput. Appl. Math., 367 (2020), 112473, 20. doi: 10.1016/j.cam.2019.112473. [22] X. He, F. Song and W. Deng, A well-conditioned, nonconforming Nitsche's extended finite element method for elliptic interface problems, Numer. Math. Theory Methods Appl., 13 (2020), 99-130.  doi: 10.4208/nmtma.OA-2019-0053. [23] H. Huang and Z. Li, Convergence analysis of the immersed interface method, IMA J. Numer. Anal., 19 (1999), 583-608.  doi: 10.1093/imanum/19.4.583. [24] P. Huang, H. Wu and Y. Xiao, An unfitted interface penalty finite element method for elliptic interface problems, Comput. Methods Appl. Mech. Engrg., 323 (2017), 439-460.  doi: 10.1016/j.cma.2017.06.004. [25] M. Kirchhart, S. Gross and A. Reusken, Analysis of an XFEM discretization for Stokes interface problems, SIAM J. Sci. Comput., 38 (2016), 1019-1043.  doi: 10.1137/15M1011779. [26] D. Y. Kwak, K. T. Wee and K. S. Chang, An analysis of broken $P_1$-nonconforming finite element method for interface problems, SIAM J. Numer. Anal., 48 (2010), 2117-2134.  doi: 10.1137/080728056. [27] Z. Li, T. Lin and X. Wu, New Cartesian grid methods for interface problems using the finite element formulation, Numer. Math., 96 (2003), 61-98. doi: 10.1007/s00211-003-0473-x. [28] T. Lin, D. Sheen and X. Zhang, A nonconforming immersed finite element method for elliptic interface problems, J. Sci. Comput., 79 (2019), 442-463.  doi: 10.1007/s10915-018-0865-9. [29] T. Lin, Q. Yang and X. Zhang, Partially penalized immersed finite element methods for parabolic interface problems, Numer. Methods Partial Differential Equations, 31 (2015), 1925-1947.  doi: 10.1002/num.21973. [30] A. Massing, M. G. Larson, A. Logg and M. E. Rognes, A stabilized Nitsche fictitious domain method for the Stokes problem, J. Sci. Comput., 61 (2014), 604-628.  doi: 10.1007/s10915-014-9838-9. [31] A. Massing, B. Schott and W. A. Wall, A stabilized Nitsche cut finite element method for the Oseen problem, Comput. Methods Appl. Mech. Engrg., 328 (2018), 262-300.  doi: 10.1016/j.cma.2017.09.003. [32] R. Massjung, An unfitted discontinuous Galerkin method applied to elliptic interface problems, SIAM J. Numer. Anal., 50 (2012), 3134-3162.  doi: 10.1137/090763093. [33] M. Sarkis, Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements, Numer. Math., 77 (1997), 383-406.  doi: 10.1007/s002110050292. [34] E. Wadbro, S. Zahedi, G. Kreiss and M. Berggren, A uniformly well-conditioned, unfitted Nitsche method for interface problems, Bit Numerical Mathematics, 53 (2013), 791-820.  doi: 10.1007/s10543-012-0417-x. [35] N. Wang and J. Chen, A nonconforming Nitsche's extended finite element method for Stokes interface problems, J. Sci. Comput., 81 (2019), 342-374.  doi: 10.1007/s10915-019-01019-9. [36] Q. Wang and J. Chen, A new unfitted stabilized Nitsche's finite element method for Stokes interface problems, Comput. Math. Appl., 70 (2015), 820-834.  doi: 10.1016/j.camwa.2015.05.024. [37] H. Wu and Y. Xiao, An unfitted $hp$-interface penalty finite element method for elliptic interface problems, J. Comput. Math., 37 (2019), 316-339.  doi: 10.4208/jcm.1802-m2017-0219. [38] Y. Xiao, J. Xu and F. Wang, High-order extended finite element methods for solving interface problems, Comput. Methods Appl. Mech. Engrg., 364 (2020), 112964, 21 pp. doi: 10.1016/j.cma.2020.112964.
A sample domain $\Omega$
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)
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)
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
 $h$ $e^1_{h,\mathbf{u}}$ 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
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
 $h$ $e^1_{h,\mathbf{u}}$ 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
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
 $\mu_1$ $\mu_2$ $e^1_{h,\mathbf{u}}$ $e^0_{h,\mathbf{u}}$ $e^0_{h,p}$ $1E+01$ $1$ 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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