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doi: 10.3934/dcdsb.2021163
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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 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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021163
References:
[1]

S. AdjeridN. 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.  Google Scholar

[2]

N. BarrauR. BeckerE. 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.  Google Scholar

[3]

R. BeckerE. 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.  Google Scholar

[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.  Google Scholar

[5]

E. Burman, Ghost penalty, C. R. Math. Acad. Sci. Paris, 348 (2010), 1217-1220.  doi: 10.1016/j.crma.2010.10.006.  Google Scholar

[6]

E. BurmanJ. GuzmánM. 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.  Google Scholar

[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.  Google Scholar

[8]

Z. CaiX. 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.  Google Scholar

[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.  Google Scholar

[10]

D. CapatinaH. EI-OtmanyD. 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.  Google Scholar

[11]

L. CattaneoL. FormaggiaG. F. IoriA. 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.  Google Scholar

[12]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.  Google Scholar

[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.  Google Scholar

[14]

A. ErnA. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[17]

Y. GongB. 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.  Google Scholar

[18]

J. GuzmánM. 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.  Google Scholar

[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.  Google Scholar

[20]

P. HansboM. 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.  Google Scholar

[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.  Google Scholar

[22]

X. HeF. 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.  Google Scholar

[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.  Google Scholar

[24]

P. HuangH. 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.  Google Scholar

[25]

M. KirchhartS. 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.  Google Scholar

[26]

D. Y. KwakK. 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.  Google Scholar

[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.  Google Scholar

[28]

T. LinD. 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.  Google Scholar

[29]

T. LinQ. 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.  Google Scholar

[30]

A. MassingM. G. LarsonA. 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.  Google Scholar

[31]

A. MassingB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

E. WadbroS. ZahediG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

S. AdjeridN. 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.  Google Scholar

[2]

N. BarrauR. BeckerE. 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.  Google Scholar

[3]

R. BeckerE. 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.  Google Scholar

[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.  Google Scholar

[5]

E. Burman, Ghost penalty, C. R. Math. Acad. Sci. Paris, 348 (2010), 1217-1220.  doi: 10.1016/j.crma.2010.10.006.  Google Scholar

[6]

E. BurmanJ. GuzmánM. 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.  Google Scholar

[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.  Google Scholar

[8]

Z. CaiX. 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.  Google Scholar

[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.  Google Scholar

[10]

D. CapatinaH. EI-OtmanyD. 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.  Google Scholar

[11]

L. CattaneoL. FormaggiaG. F. IoriA. 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.  Google Scholar

[12]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.  Google Scholar

[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.  Google Scholar

[14]

A. ErnA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

Y. GongB. 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.  Google Scholar

[18]

J. GuzmánM. 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.  Google Scholar

[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.  Google Scholar

[20]

P. HansboM. 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.  Google Scholar

[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.  Google Scholar

[22]

X. HeF. 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.  Google Scholar

[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.  Google Scholar

[24]

P. HuangH. 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.  Google Scholar

[25]

M. KirchhartS. 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.  Google Scholar

[26]

D. Y. KwakK. 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.  Google Scholar

[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.  Google Scholar

[28]

T. LinD. 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.  Google Scholar

[29]

T. LinQ. 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.  Google Scholar

[30]

A. MassingM. G. LarsonA. 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.  Google Scholar

[31]

A. MassingB. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

E. WadbroS. ZahediG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  A sample domain $ \Omega $
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)
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)
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
$ 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
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
$ 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
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
$ \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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