# American Institute of Mathematical Sciences

November  2021, 29(5): 3171-3191. doi: 10.3934/era.2021032

## A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces

 1 Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA 2 Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA

* Corresponding author

Received  November 2020 Revised  February 2021 Published  November 2021 Early access  April 2021

Fund Project: The first author is supported by NSF Graduate Research Fellowship NO. 1645630. The second author is supported by NSF Grants DMS-1720425 and DMS-2005272

In this article, we develop a new mixed immersed finite element discretization for two-dimensional unsteady Stokes interface problems with unfitted meshes. The proposed IFE spaces use conforming linear elements for one velocity component and non-conforming linear elements for the other velocity component. The pressure is approximated by piecewise constant. Unisolvency, among other fundamental properties of the new vector-valued IFE functions, is analyzed. Based on the new IFE spaces, semi-discrete and full-discrete schemes are developed for solving the unsteady Stokes equations with a stationary or a moving interface. Re-meshing is not required in our numerical scheme for solving the moving-interface problem. Numerical experiments are carried out to demonstrate the performance of this new IFE method.

Citation: Derrick Jones, Xu Zhang. A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces. Electronic Research Archive, 2021, 29 (5) : 3171-3191. doi: 10.3934/era.2021032
##### 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.  Google Scholar [2] D. N. Arnold, On nonconforming linear-constant elements for some variants of the Stokes equations, Istit. Lombardo Accad. Sci. Lett. Rend. A, 127 (1993), 83-93.   Google Scholar [3] N. Chaabane, Immersed and Discontinuous Finite Element Methods, Thesis (Ph.D.)-Virginia Polytechnic Institute and State University. 2015.  Google Scholar [4] Z. Chen, Finite Element Methods and their Applications, Scientific Computation. Springer-Verlag, Berlin, 2005. Google Scholar [5] Y. Chen and X. Zhang, A $P_2$-$P_1$ partially penalized immersed finite element method for Stokes interface problems, Int. J. Numer. Anal. Model., 18 (2021), 120-141.   Google Scholar [6] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Franç caise Automat. Informat. Recherche Opérationnelle Sér. Rouge, 7 (1973), 33-75.   Google Scholar [7] F. Duarte, R. Gormaz and S. Natesan, Arbitrary Lagrangian-Eulerian method for Navier-Stokes equations with moving boundaries, Comput. Methods Appl. Mech. Engrg., 193 (2004), 4819-4836.  doi: 10.1016/j.cma.2004.05.003.  Google Scholar [8] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, volume 5 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1986. Theory and algorithms. doi: 10.1007/978-3-642-61623-5.  Google Scholar [9] S. Großand and A. Reusken, An extended pressure finite element space for two-phase incompressible flows with surface tension, J. Comput. Phys., 224 (2007), 40-58.  doi: 10.1016/j.jcp.2006.12.021.  Google Scholar [10] R. Guo, Solving parabolic moving interface problems with dynamical immersed spaces on unfitted meshes: Fully discrete analysis, SIAM J. Numer. Anal., 59 (2021), 797-828.  doi: 10.1137/20M133508X.  Google Scholar [11] R. Guo and T. Lin, A group of immersed finite element spaces for elliptic interface problems, IMA J. Numer. Anal., 39 (2019), 482-511.  doi: 10.1093/imanum/drx074.  Google Scholar [12] R. Guo, T. Lin and Y. Lin, A fixed mesh method with immersed finite elements for solving interface inverse problems, J. Sci. Comput., 79 (2019), 148-175.  doi: 10.1007/s10915-018-0847-y.  Google Scholar [13] R. Guo, T. Lin and Y. Lin, Recovering elastic inclusions by shape optimization methods with immersed finite elements, J. Comput. Phys., 404 (2020), 109123, 24 pp. doi: 10.1016/j.jcp.2019.109123.  Google Scholar [14] R. Guo, T. Lin and Q. Zhuang, Improved error estimation for the partially penalized immersed finite element methods for elliptic interface problems, Int. J. Numer. Anal. 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Chen, A new local stabilized nonconforming finite element method for the Stokes equations, Computing, 82 (2008), 157-170.  doi: 10.1007/s00607-008-0001-z.  Google Scholar [24] 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 [25] T. Lin, Y. Lin and X. Zhang, A method of lines based on immersed finite elements for parabolic moving interface problems, Adv. Appl. Math. Mech., 5 (2013), 548-568.  doi: 10.4208/aamm.13-13S11.  Google Scholar [26] T. Lin, Y. Lin and X. Zhang, Partially penalized immersed finite element methods for elliptic interface problems, SIAM J. Numer. Anal., 53 (2015), 1121-1144.  doi: 10.1137/130912700.  Google Scholar [27] T. Lin, D. Sheen and X. Zhang, A locking-free immersed finite element method for planar elasticity interface problems, J. Comput. Phys., 247 (2013), 228-247.  doi: 10.1016/j.jcp.2013.03.053.  Google Scholar [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.  Google Scholar [29] T. Lin and X. Zhang, Linear and bilinear immersed finite elements for planar elasticity interface problems, J. Comput. Appl. Math., 236 (2012), 4681-4699.  doi: 10.1016/j.cam.2012.03.012.  Google Scholar [30] T. Lin and Q. Zhuang, Optimal error bounds for partially penalized immersed finite element methods for parabolic interface problems, J. Comput. Appl. Math., 366 (2020), 112401, 11 pp. doi: 10.1016/j.cam.2019.112401.  Google Scholar [31] A. Lundberg, P. Sun and C. Wang, Distributed Lagrange multiplier-fictitious domain finite element method for Stokes interface problems, Int. J. Numer. Anal. Model., 16 (2019), 939-963.   Google Scholar [32] R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numer. Methods Partial Differential Equations, 8 (1992), 97-111.  doi: 10.1002/num.1690080202.  Google Scholar [33] B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, volume 35 of Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. Theory and implementation. doi: 10.1137/1.9780898717440.  Google Scholar [34] P. Sun, Fictitious domain finite element method for Stokes/elliptic interface problems with jump coefficients, J. Comput. Appl. Math., 356 (2019), 81-97.  doi: 10.1016/j.cam.2019.01.030.  Google Scholar [35] C. Taylor and P. Hood, A numerical solution of the Navier-Stokes equations using the finite element technique, Internat. J. Comput. & Fluids, 1 (1973), 73-100.  doi: 10.1016/0045-7930(73)90027-3.  Google Scholar [36] 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 [37] N. K. Yamaleev, D. C. Del Rey Fernández, J. Lou and M. H. Carpenter, Entropy stable spectral collocation schemes for the 3-D Navier-Stokes equations on dynamic unstructured grids, J. Comput. Phys., 399 (2019), 108897, 27 pp. doi: 10.1016/j.jcp.2019.108897.  Google Scholar [38] M. Zhang and S. Zhang, A 3D conforming-nonconforming mixed finite element for solving symmetric stress Stokes equations, Int. J. Numer. Anal. Model., 14 (2017), 730-743.   Google Scholar

show all references

##### 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.  Google Scholar [2] D. N. Arnold, On nonconforming linear-constant elements for some variants of the Stokes equations, Istit. Lombardo Accad. Sci. Lett. Rend. A, 127 (1993), 83-93.   Google Scholar [3] N. Chaabane, Immersed and Discontinuous Finite Element Methods, Thesis (Ph.D.)-Virginia Polytechnic Institute and State University. 2015.  Google Scholar [4] Z. Chen, Finite Element Methods and their Applications, Scientific Computation. Springer-Verlag, Berlin, 2005. Google Scholar [5] Y. Chen and X. Zhang, A $P_2$-$P_1$ partially penalized immersed finite element method for Stokes interface problems, Int. J. Numer. Anal. Model., 18 (2021), 120-141.   Google Scholar [6] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Franç caise Automat. Informat. Recherche Opérationnelle Sér. Rouge, 7 (1973), 33-75.   Google Scholar [7] F. Duarte, R. Gormaz and S. Natesan, Arbitrary Lagrangian-Eulerian method for Navier-Stokes equations with moving boundaries, Comput. Methods Appl. Mech. Engrg., 193 (2004), 4819-4836.  doi: 10.1016/j.cma.2004.05.003.  Google Scholar [8] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, volume 5 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1986. Theory and algorithms. doi: 10.1007/978-3-642-61623-5.  Google Scholar [9] S. Großand and A. Reusken, An extended pressure finite element space for two-phase incompressible flows with surface tension, J. Comput. Phys., 224 (2007), 40-58.  doi: 10.1016/j.jcp.2006.12.021.  Google Scholar [10] R. Guo, Solving parabolic moving interface problems with dynamical immersed spaces on unfitted meshes: Fully discrete analysis, SIAM J. Numer. Anal., 59 (2021), 797-828.  doi: 10.1137/20M133508X.  Google Scholar [11] R. Guo and T. Lin, A group of immersed finite element spaces for elliptic interface problems, IMA J. Numer. Anal., 39 (2019), 482-511.  doi: 10.1093/imanum/drx074.  Google Scholar [12] R. Guo, T. Lin and Y. Lin, A fixed mesh method with immersed finite elements for solving interface inverse problems, J. Sci. Comput., 79 (2019), 148-175.  doi: 10.1007/s10915-018-0847-y.  Google Scholar [13] R. Guo, T. Lin and Y. Lin, Recovering elastic inclusions by shape optimization methods with immersed finite elements, J. Comput. Phys., 404 (2020), 109123, 24 pp. doi: 10.1016/j.jcp.2019.109123.  Google Scholar [14] R. Guo, T. Lin and Q. Zhuang, Improved error estimation for the partially penalized immersed finite element methods for elliptic interface problems, Int. J. Numer. Anal. Model., 16 (2019), 575-589.   Google Scholar [15] 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.  Google Scholar [16] X. He, T. Lin and Y. Lin, Immersed finite element methods for elliptic interface problems with non-homogeneous jump conditions, Int. J. Numer. Anal. Model., 8 (2011), 284-301.   Google Scholar [17] X. He, T. Lin, Y. Lin and X. Zhang, Immersed finite element methods for parabolic equations with moving interface, Numer. Methods Partial Differential Equations, 29 (2013), 619-646.  doi: 10.1002/num.21722.  Google Scholar [18] C. He and X. Zhang, Residual-based a posteriori error estimation for immersed finite element methods, J. Sci. Comput., 81 (2019), 2051-2079.  doi: 10.1007/s10915-019-01071-5.  Google Scholar [19] V. John, Finite Element Methods for Incompressible Flow Problems, volume 51 of Springer Series in Computational Mathematics, Springer, Cham, 2016. doi: 10.1007/978-3-319-45750-5.  Google Scholar [20] D. Jones and X. Zhang, A class of nonconforming immersed finite element methods for Stokes interface problems, J. Comput. Appl. Math., 392 (2021), 113493. doi: 10.1016/j.cam.2021.113493.  Google Scholar [21] R. Kouhia and R. Stenberg, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow, Comput. Methods Appl. Mech. Engrg., 124 (1995), 195-212.  doi: 10.1016/0045-7825(95)00829-P.  Google Scholar [22] R. Lan and P. Sun, A monolithic arbitrary Lagrangian-Eulerian finite element analysis for a Stokes/parabolic moving interface problem, J. Sci. Comput., 82 (2020), Paper No. 59, 36 pp. doi: 10.1007/s10915-020-01161-9.  Google Scholar [23] J. Li and Z. Chen, A new local stabilized nonconforming finite element method for the Stokes equations, Computing, 82 (2008), 157-170.  doi: 10.1007/s00607-008-0001-z.  Google Scholar [24] 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 [25] T. Lin, Y. Lin and X. Zhang, A method of lines based on immersed finite elements for parabolic moving interface problems, Adv. Appl. Math. Mech., 5 (2013), 548-568.  doi: 10.4208/aamm.13-13S11.  Google Scholar [26] T. Lin, Y. Lin and X. Zhang, Partially penalized immersed finite element methods for elliptic interface problems, SIAM J. Numer. Anal., 53 (2015), 1121-1144.  doi: 10.1137/130912700.  Google Scholar [27] T. Lin, D. Sheen and X. Zhang, A locking-free immersed finite element method for planar elasticity interface problems, J. Comput. Phys., 247 (2013), 228-247.  doi: 10.1016/j.jcp.2013.03.053.  Google Scholar [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.  Google Scholar [29] T. Lin and X. Zhang, Linear and bilinear immersed finite elements for planar elasticity interface problems, J. Comput. Appl. Math., 236 (2012), 4681-4699.  doi: 10.1016/j.cam.2012.03.012.  Google Scholar [30] T. Lin and Q. Zhuang, Optimal error bounds for partially penalized immersed finite element methods for parabolic interface problems, J. Comput. Appl. Math., 366 (2020), 112401, 11 pp. doi: 10.1016/j.cam.2019.112401.  Google Scholar [31] A. Lundberg, P. Sun and C. Wang, Distributed Lagrange multiplier-fictitious domain finite element method for Stokes interface problems, Int. J. Numer. Anal. Model., 16 (2019), 939-963.   Google Scholar [32] R. Rannacher and S. Turek, Simple nonconforming quadrilateral Stokes element, Numer. Methods Partial Differential Equations, 8 (1992), 97-111.  doi: 10.1002/num.1690080202.  Google Scholar [33] B. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, volume 35 of Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2008. Theory and implementation. doi: 10.1137/1.9780898717440.  Google Scholar [34] P. Sun, Fictitious domain finite element method for Stokes/elliptic interface problems with jump coefficients, J. Comput. Appl. Math., 356 (2019), 81-97.  doi: 10.1016/j.cam.2019.01.030.  Google Scholar [35] C. Taylor and P. Hood, A numerical solution of the Navier-Stokes equations using the finite element technique, Internat. J. Comput. & Fluids, 1 (1973), 73-100.  doi: 10.1016/0045-7930(73)90027-3.  Google Scholar [36] 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 [37] N. K. Yamaleev, D. C. Del Rey Fernández, J. Lou and M. H. Carpenter, Entropy stable spectral collocation schemes for the 3-D Navier-Stokes equations on dynamic unstructured grids, J. Comput. Phys., 399 (2019), 108897, 27 pp. doi: 10.1016/j.jcp.2019.108897.  Google Scholar [38] M. Zhang and S. Zhang, A 3D conforming-nonconforming mixed finite element for solving symmetric stress Stokes equations, Int. J. Numer. Anal. Model., 14 (2017), 730-743.   Google Scholar
The geometrical setup of a moving interface problem
From left: an interface-fitted mesh and an unfitted mesh
Types of interface elements. From left: Type Ⅰ, Type Ⅱ, Type Ⅲ
A comparison of the vector-valued IFE shape function $\mathit{\boldsymbol{\phi}}_{4, T}$ with $\mu^- = 1$, $\mu^+ = 5$ (top), and the corresponding FE shape function $\mathit{\boldsymbol{\psi}}_{4, T}$ (bottom) on the reference triangle
An illustration of a moving interface in two consecutive steps. Elements in dark yellow indicate interface configuration changes, and elements in dark blue remain unchanged
CR-$P_1$-$P_0$ IFE Solution of Example 5.3 with $\mu^- = 1$ and $\mu^+ = 10$ on the $64\times 64$ mesh at times $t = 0.25$, $0.75$, and $1$. Top plots: Interfaces, middle: IFE solutions $u_{1h}$, bottom: IFE solutions $u_{2h}$
CR-$P_1$-$P_0$ IFE Interpolation errors for Example 5.1 with $\mu^- = 1$ and $\mu^+ = 10$
 N $e^0(u_{1, I})$ rate $e^0(u_{2, I})$ rate $e^0(p_I)$ rate $e^1(u_{1, I})$ rate $e^1(u_{2, I})$ rate $8$ 5.36e-3 n/a 1.15e-2 n/a 7.02e-2 n/a 1.21e-1 n/a 1.54e-1 n/a $16$ 1.39e-3 1.95 3.03e-3 1.92 3.14e-2 1.16 5.80e-2 1.06 7.32e-2 1.06 $32$ 3.59e-4 1.95 7.84e-4 1.95 1.46e-2 1.10 2.85e-2 1.02 3.73e-2 0.96 $64$ 9.20e-5 1.96 2.03e-4 1.95 5.28e-3 1.47 1.45e-2 0.98 1.91e-2 0.97 $128$ 2.33e-5 1.98 5.14e-5 1.98 2.10e-3 1.33 7.34e-3 0.98 9.66e-3 0.98 $256$ 5.85e-6 1.99 1.29e-5 1.99 8.47e-4 1.31 3.68e-3 1.00 4.85e-3 0.99 rate 1.98 1.96 1.29 1.00 0.99
 N $e^0(u_{1, I})$ rate $e^0(u_{2, I})$ rate $e^0(p_I)$ rate $e^1(u_{1, I})$ rate $e^1(u_{2, I})$ rate $8$ 5.36e-3 n/a 1.15e-2 n/a 7.02e-2 n/a 1.21e-1 n/a 1.54e-1 n/a $16$ 1.39e-3 1.95 3.03e-3 1.92 3.14e-2 1.16 5.80e-2 1.06 7.32e-2 1.06 $32$ 3.59e-4 1.95 7.84e-4 1.95 1.46e-2 1.10 2.85e-2 1.02 3.73e-2 0.96 $64$ 9.20e-5 1.96 2.03e-4 1.95 5.28e-3 1.47 1.45e-2 0.98 1.91e-2 0.97 $128$ 2.33e-5 1.98 5.14e-5 1.98 2.10e-3 1.33 7.34e-3 0.98 9.66e-3 0.98 $256$ 5.85e-6 1.99 1.29e-5 1.99 8.47e-4 1.31 3.68e-3 1.00 4.85e-3 0.99 rate 1.98 1.96 1.29 1.00 0.99
$P_1$-CR-$P_0$ IFE Interpolation errors for Example 5.1 with $\mu^- = 1$ and $\mu^+ = 10$
 N $e^0(u_{1, I})$ rate $e^0(u_{2, I})$ rate $e^0(p_I)$ rate $e^1(u_{1, I})$ rate $e^1(u_{2, I})$ rate $8$ 1.16e-2 n/a 5.44e-3 n/a 1.44e-1 n/a 1.49e-1 n/a 1.30e-1 n/a $16$ 3.08e-3 1.92 1.42e-3 1.94 5.93e-2 1.29 7.47e-2 1.00 5.80e-2 1.16 $32$ 5.15e-4 1.95 2.36e-4 1.96 2.14e-2 1.18 3.08e-2 0.96 2.37e-2 0.98 $64$ 7.94e-4 1.96 3.65e-4 1.97 2.70e-2 1.14 3.76e-2 0.99 2.88e-2 1.00 $128$ 5.15e-5 1.99 2.34e-5 1.99 3.56e-3 1.43 9.69e-3 0.98 7.35e-3 0.99 $256$ 1.29e-5 1.99 5.86e-6 2.00 1.32e-3 1.43 4.86e-3 0.99 3.68e-3 1.00 rate 1.89 1.90 1.31 0.95 0.98
 N $e^0(u_{1, I})$ rate $e^0(u_{2, I})$ rate $e^0(p_I)$ rate $e^1(u_{1, I})$ rate $e^1(u_{2, I})$ rate $8$ 1.16e-2 n/a 5.44e-3 n/a 1.44e-1 n/a 1.49e-1 n/a 1.30e-1 n/a $16$ 3.08e-3 1.92 1.42e-3 1.94 5.93e-2 1.29 7.47e-2 1.00 5.80e-2 1.16 $32$ 5.15e-4 1.95 2.36e-4 1.96 2.14e-2 1.18 3.08e-2 0.96 2.37e-2 0.98 $64$ 7.94e-4 1.96 3.65e-4 1.97 2.70e-2 1.14 3.76e-2 0.99 2.88e-2 1.00 $128$ 5.15e-5 1.99 2.34e-5 1.99 3.56e-3 1.43 9.69e-3 0.98 7.35e-3 0.99 $256$ 1.29e-5 1.99 5.86e-6 2.00 1.32e-3 1.43 4.86e-3 0.99 3.68e-3 1.00 rate 1.89 1.90 1.31 0.95 0.98
$P_1$-CR-$P_0$ IFE Interpolation errors for Example 5.1 with $\mu^- = 1$ and $\mu^+ = 200$
 N $e^0(u_{1, I})$ rate $e^0(u_{2, I})$ rate $e^0(p_I)$ rate $e^1(u_{1, I})$ rate $e^1(u_{2, I})$ rate $8$ 1.01e-2 n/a 4.86e-2 n/a 2.81e-0 n/a 1.35e-1 n/a 1.26e-1 n/a $16$ 2.73e-3 1.88 1.28e-3 1.92 1.21e-0 1.21 6.77e-2 1.00 5.31e-2 1.24 $32$ 7.19e-4 1.93 3.33e-4 1.95 5.75e-1 1.08 3.43e-2 0.98 2.66e-2 1.00 $64$ 1.86e-4 1.95 8.59e-5 1.97 1.98e-2 1.54 1.75e-2 0.97 1.34e-2 0.99 $128$ 4.73e-5 1.98 2.15e-5 1.98 7.26e-2 1.45 8.91e-3 0.98 6.79e-3 0.98 $256$ 1.19e-5 1.99 5.40e-6 1.99 2.59e-2 1.49 4.49e-3 0.99 3.41e-3 0.99 rate 1.95 1.90 1.45 0.98 1.03
 N $e^0(u_{1, I})$ rate $e^0(u_{2, I})$ rate $e^0(p_I)$ rate $e^1(u_{1, I})$ rate $e^1(u_{2, I})$ rate $8$ 1.01e-2 n/a 4.86e-2 n/a 2.81e-0 n/a 1.35e-1 n/a 1.26e-1 n/a $16$ 2.73e-3 1.88 1.28e-3 1.92 1.21e-0 1.21 6.77e-2 1.00 5.31e-2 1.24 $32$ 7.19e-4 1.93 3.33e-4 1.95 5.75e-1 1.08 3.43e-2 0.98 2.66e-2 1.00 $64$ 1.86e-4 1.95 8.59e-5 1.97 1.98e-2 1.54 1.75e-2 0.97 1.34e-2 0.99 $128$ 4.73e-5 1.98 2.15e-5 1.98 7.26e-2 1.45 8.91e-3 0.98 6.79e-3 0.98 $256$ 1.19e-5 1.99 5.40e-6 1.99 2.59e-2 1.49 4.49e-3 0.99 3.41e-3 0.99 rate 1.95 1.90 1.45 0.98 1.03
$P_1$-CR-$P_0$ IFE Interpolation errors for Example 5.1 with $\mu^- = 10$ and $\mu^+ = 1$
 $N_s$ $e^0(u_{1, I})$ rate $e^0(u_{2, I})$ rate $e^0(p_I)$ rate $e^1(u_{1, I})$ rate $e^1(u_{2, I})$ rate $8$ 5.11e-2 n/a 2.32e-2 n/a 3.38e-1 n/a 6.04e-1 n/a 4.61e-1 n/a $16$ 1.29e-2 1.99 5.82e-3 1.99 9.59e-2 1.82 3.02e-1 1.00 2.29e-1 1.01 $32$ 3.23e-3 1.99 1.46e-3 2.00 2.36e-2 2.03 1.51e-1 1.00 1.15e-1 1.00 $64$ 8.09e-4 2.00 3.66e-4 2.00 1.07e-2 1.14 7.58e-2 1.00 5.73e-2 1.00 $128$ 2.02e-4 2.00 9.14e-5 2.00 3.41e-3 1.65 3.79e-2 1.00 2.87e-2 1.00 $256$ 5.06e-5 2.00 2.29e-5 2.00 1.37e-3 1.32 1.90e-2 1.00 1.43e-2 1.00 rate 2.00 2.00 1.58 1.00 1.00
 $N_s$ $e^0(u_{1, I})$ rate $e^0(u_{2, I})$ rate $e^0(p_I)$ rate $e^1(u_{1, I})$ rate $e^1(u_{2, I})$ rate $8$ 5.11e-2 n/a 2.32e-2 n/a 3.38e-1 n/a 6.04e-1 n/a 4.61e-1 n/a $16$ 1.29e-2 1.99 5.82e-3 1.99 9.59e-2 1.82 3.02e-1 1.00 2.29e-1 1.01 $32$ 3.23e-3 1.99 1.46e-3 2.00 2.36e-2 2.03 1.51e-1 1.00 1.15e-1 1.00 $64$ 8.09e-4 2.00 3.66e-4 2.00 1.07e-2 1.14 7.58e-2 1.00 5.73e-2 1.00 $128$ 2.02e-4 2.00 9.14e-5 2.00 3.41e-3 1.65 3.79e-2 1.00 2.87e-2 1.00 $256$ 5.06e-5 2.00 2.29e-5 2.00 1.37e-3 1.32 1.90e-2 1.00 1.43e-2 1.00 rate 2.00 2.00 1.58 1.00 1.00
$P_1$-CR-$P_0$ backward-Euler IFE solutions for Example 5.2 at $t = 1$ with $\mu^- = 1$ and $\mu^+ = 10$
 N $e^0(u_{1, I})$ rate $e^0(u_{2, I})$ rate $e^0(p_I)$ rate $e^1(u_{1, I})$ rate $e^1(u_{2, I})$ rate $8$ 2.49e-1 n/a 1.72e-1 n/a 9.46e-0 n/a 2.95e-0 n/a 2.83e-0 n/a $16$ 6.86e-2 1.86 4.70e-2 1.87 4.70e-0 1.01 1.51e-0 0.97 1.38e-0 1.03 $32$ 1.69e-2 2.02 1.18e-2 1.99 2.44e-0 0.95 7.65e-1 0.98 7.14e-1 0.96 $64$ 3.87e-3 2.13 3.54e-3 1.74 1.15e-0 1.08 3.94e-1 0.96 3.69e-1 0.95 $128$ 1.57e-3 1.31 1.65e-3 1.10 6.23e-1 0.88 2.04e-1 0.95 1.91e-1 0.95 $256$ 8.69e-4 0.85 9.07e-4 0.86 3.35e-1 0.90 1.07e-1 0.93 1.02e-1 0.91 rate 1.69 1.54 0.97 0.96 0.96
 N $e^0(u_{1, I})$ rate $e^0(u_{2, I})$ rate $e^0(p_I)$ rate $e^1(u_{1, I})$ rate $e^1(u_{2, I})$ rate $8$ 2.49e-1 n/a 1.72e-1 n/a 9.46e-0 n/a 2.95e-0 n/a 2.83e-0 n/a $16$ 6.86e-2 1.86 4.70e-2 1.87 4.70e-0 1.01 1.51e-0 0.97 1.38e-0 1.03 $32$ 1.69e-2 2.02 1.18e-2 1.99 2.44e-0 0.95 7.65e-1 0.98 7.14e-1 0.96 $64$ 3.87e-3 2.13 3.54e-3 1.74 1.15e-0 1.08 3.94e-1 0.96 3.69e-1 0.95 $128$ 1.57e-3 1.31 1.65e-3 1.10 6.23e-1 0.88 2.04e-1 0.95 1.91e-1 0.95 $256$ 8.69e-4 0.85 9.07e-4 0.86 3.35e-1 0.90 1.07e-1 0.93 1.02e-1 0.91 rate 1.69 1.54 0.97 0.96 0.96
$P_1$-CR-$P_0$ Crank-Nicolson IFE solutions for Example 5.2 at $t = 1$ with $\mu^- = 1$ and $\mu^+ = 10$
 N $e^0(u_{1, I})$ rate $e^0(u_{2, I})$ rate $e^0(p_I)$ rate $e^1(u_{1, I})$ rate $e^1(u_{2, I})$ rate $8$ 2.51e-1 n/a 1.72e-1 n/a 9.02e-0 n/a 2.94e-0 n/a 2.79e-0 n/a $16$ 7.25e-2 1.79 5.02e-2 1.77 4.51e-0 1.00 1.50e-0 0.97 1.36e-0 1.04 $32$ 1.92e-2 1.92 1.39e-2 1.85 2.34e-0 0.94 7.62e-1 0.98 6.98e-1 0.96 $64$ 4.33e-3 2.15 3.27e-3 2.09 1.11e-0 1.08 3.92e-1 0.96 3.61e-1 0.95 $128$ 9.96e-4 2.12 7.94e-4 2.04 5.97e-1 0.89 2.03e-1 0.95 1.87e-1 0.95 $256$ 2.39e-4 2.06 2.33e-4 1.76 3.20e-1 0.90 1.06e-1 0.93 1.02e-1 0.91 rate 2.03 1.93 0.97 0.96 0.96
 N $e^0(u_{1, I})$ rate $e^0(u_{2, I})$ rate $e^0(p_I)$ rate $e^1(u_{1, I})$ rate $e^1(u_{2, I})$ rate $8$ 2.51e-1 n/a 1.72e-1 n/a 9.02e-0 n/a 2.94e-0 n/a 2.79e-0 n/a $16$ 7.25e-2 1.79 5.02e-2 1.77 4.51e-0 1.00 1.50e-0 0.97 1.36e-0 1.04 $32$ 1.92e-2 1.92 1.39e-2 1.85 2.34e-0 0.94 7.62e-1 0.98 6.98e-1 0.96 $64$ 4.33e-3 2.15 3.27e-3 2.09 1.11e-0 1.08 3.92e-1 0.96 3.61e-1 0.95 $128$ 9.96e-4 2.12 7.94e-4 2.04 5.97e-1 0.89 2.03e-1 0.95 1.87e-1 0.95 $256$ 2.39e-4 2.06 2.33e-4 1.76 3.20e-1 0.90 1.06e-1 0.93 1.02e-1 0.91 rate 2.03 1.93 0.97 0.96 0.96
CR-$P_1$-$P_0$ Backward-Euler IFE solution for Example 5.3 at $t = 1$ with $\mu^- = 1$ and $\mu^+ = 10$
 N $e^0(u_{1, I})$ rate $e^0(u_{2, I})$ rate $e^0(p_I)$ rate $e^1(u_{1, I})$ rate $e^1(u_{2, I})$ rate $8$ 7.85e-3 n/a 1.14e-2 n/a 4.83e-1 n/a 1.36e-1 n/a 1.51e-1 n/a $16$ 2.05e-3 1.94 2.95e-3 1.95 2.41e-1 1.00 7.02e-2 0.95 7.45e-2 1.02 $32$ 5.13e-4 2.00 6.54e-4 2.17 1.24e-1 0.96 3.57e-2 0.98 3.82e-2 0.96 $64$ 1.68e-4 1.61 1.32e-4 2.30 5.78e-2 1.10 1.84e-2 0.96 1.96e-2 0.96 $128$ 8.54e-5 0.98 6.68e-5 0.99 3.12e-2 0.89 9.52e-3 0.95 1.01e-2 0.95 rate 1.67 1.93 1.00 0.96 0.97
 N $e^0(u_{1, I})$ rate $e^0(u_{2, I})$ rate $e^0(p_I)$ rate $e^1(u_{1, I})$ rate $e^1(u_{2, I})$ rate $8$ 7.85e-3 n/a 1.14e-2 n/a 4.83e-1 n/a 1.36e-1 n/a 1.51e-1 n/a $16$ 2.05e-3 1.94 2.95e-3 1.95 2.41e-1 1.00 7.02e-2 0.95 7.45e-2 1.02 $32$ 5.13e-4 2.00 6.54e-4 2.17 1.24e-1 0.96 3.57e-2 0.98 3.82e-2 0.96 $64$ 1.68e-4 1.61 1.32e-4 2.30 5.78e-2 1.10 1.84e-2 0.96 1.96e-2 0.96 $128$ 8.54e-5 0.98 6.68e-5 0.99 3.12e-2 0.89 9.52e-3 0.95 1.01e-2 0.95 rate 1.67 1.93 1.00 0.96 0.97
CR-$P_1$-$P_0$ Backward-Euler IFE solution for Example 5.3 at $t = 1$ with $\mu^- = 1$ and $\mu^+ = 200$
 N $e^0(u_{1, I})$ rate $e^0(u_{2, I})$ rate $e^0(p_I)$ rate $e^1(u_{1, I})$ rate $e^1(u_{2, I})$ rate $8$ 1.17e-2 n/a 1.29e-2 n/a 1.25e-0 n/a 1.44e-1 n/a 1.41e-1 n/a $16$ 3.86e-3 1.60 4.56e-3 1.50 8.16e-1 0.61 7.99e-2 0.85 7.01e-1 1.01 $32$ 1.20e-3 1.69 1.42e-3 1.69 5.10e-1 0.68 3.80e-2 1.07 3.56e-2 0.98 $64$ 2.02e-4 2.57 2.50e-4 2.50 2.00e-1 1.35 1.74e-2 1.12 1.78e-2 1.00 $128$ 3.48e-5 2.54 4.21e-5 2.57 8.70e-2 1.20 8.43e-3 1.05 9.01e-3 0.98 rate 2.10 2.07 0.97 1.04 1.04
 N $e^0(u_{1, I})$ rate $e^0(u_{2, I})$ rate $e^0(p_I)$ rate $e^1(u_{1, I})$ rate $e^1(u_{2, I})$ rate $8$ 1.17e-2 n/a 1.29e-2 n/a 1.25e-0 n/a 1.44e-1 n/a 1.41e-1 n/a $16$ 3.86e-3 1.60 4.56e-3 1.50 8.16e-1 0.61 7.99e-2 0.85 7.01e-1 1.01 $32$ 1.20e-3 1.69 1.42e-3 1.69 5.10e-1 0.68 3.80e-2 1.07 3.56e-2 0.98 $64$ 2.02e-4 2.57 2.50e-4 2.50 2.00e-1 1.35 1.74e-2 1.12 1.78e-2 1.00 $128$ 3.48e-5 2.54 4.21e-5 2.57 8.70e-2 1.20 8.43e-3 1.05 9.01e-3 0.98 rate 2.10 2.07 0.97 1.04 1.04
Condition Number for Backward-Euler CR-$P_1$-$P_0$ Example 5.3 with $\mu^- = 1$
 $N_s$ $\mu^+=0.01$ $\mu^+=0.1$ $\mu^+=1$ $\mu^+=10$ $\mu^+=100$ t=0.25 $8$ 3.03e+05 5.94e+04 2.80e+05 1.38e+07 1.31e+09 $16$ 1.04e+06 7.82e+05 4.36e+06 1.11e+08 1.40e+10 $32$ 2.69e+08 6.06e+06 6.87e+07 9.06e+08 4.64e+11 $64$ 6.51e+10 7.07e+07 1.09e+09 8.46e+09 8.48e+12 $128$ 1.30e+12 7.27e+08 1.74e+10 8.15e+10 6.26e+14 t=0.75 $8$ 2.07e+04 4.24e+04 2.80e+05 1.22e+07 1.78e+09 $16$ 1.04e+06 7.82e+05 4.36e+06 1.64e+08 2.22e+10 $32$ 1.15e+08 9.36e+06 6.87e+07 1.67e+09 1.79e+11 $64$ 2.44e+09 1.11e+08 1.09e+09 1.62e+10 7.07e+13 $128$ 1.22e+10 9.29e+08 1.74e+10 1.16e+11 2.66e+15 t=1 $8$ 2.34e+06 3.68e+04 2.80e+05 1.26e+07 1.10e+09 $16$ 7.76e+06 5.65e+05 4.36e+06 1.08e+08 1.94e+10 $32$ 4.30e+07 8.53e+06 6.87e+07 1.41e+09 1.59e+13 $64$ 2.99e+08 9.10e+07 1.09e+09 1.05e+10 3.93e+13 $128$ 7.30e+11 8.24e+08 1.74e+10 9.94e+10 2.72e+15
 $N_s$ $\mu^+=0.01$ $\mu^+=0.1$ $\mu^+=1$ $\mu^+=10$ $\mu^+=100$ t=0.25 $8$ 3.03e+05 5.94e+04 2.80e+05 1.38e+07 1.31e+09 $16$ 1.04e+06 7.82e+05 4.36e+06 1.11e+08 1.40e+10 $32$ 2.69e+08 6.06e+06 6.87e+07 9.06e+08 4.64e+11 $64$ 6.51e+10 7.07e+07 1.09e+09 8.46e+09 8.48e+12 $128$ 1.30e+12 7.27e+08 1.74e+10 8.15e+10 6.26e+14 t=0.75 $8$ 2.07e+04 4.24e+04 2.80e+05 1.22e+07 1.78e+09 $16$ 1.04e+06 7.82e+05 4.36e+06 1.64e+08 2.22e+10 $32$ 1.15e+08 9.36e+06 6.87e+07 1.67e+09 1.79e+11 $64$ 2.44e+09 1.11e+08 1.09e+09 1.62e+10 7.07e+13 $128$ 1.22e+10 9.29e+08 1.74e+10 1.16e+11 2.66e+15 t=1 $8$ 2.34e+06 3.68e+04 2.80e+05 1.26e+07 1.10e+09 $16$ 7.76e+06 5.65e+05 4.36e+06 1.08e+08 1.94e+10 $32$ 4.30e+07 8.53e+06 6.87e+07 1.41e+09 1.59e+13 $64$ 2.99e+08 9.10e+07 1.09e+09 1.05e+10 3.93e+13 $128$ 7.30e+11 8.24e+08 1.74e+10 9.94e+10 2.72e+15
Condition Number for Crank-Nicolson CR-$P_1$-$P_0$ Example 5.3 with $\mu^- = 1$
 $N_s$ $\mu^+=0.01$ $\mu^+=0.1$ $\mu^+=1$ $\mu^+=10$ $\mu^+=100$ t=0.25 $8$ 4.92e+05 7.56e+04 2.88e+05 1.39e+07 1.32e+09 $16$ 1.43e+06 9.20e+05 4.42e+06 1.12e+08 1.40e+10 $32$ 3.29e+08 6.58e+06 6.92e+07 9.08e+08 4.65e+11 $64$ 7.54e+10 7.37e+07 1.10e+09 8.48e+09 8.48e+12 $128$ 1.43e+12 7.42e+08 1.74e+10 8.16e+10 6.26e+14 t=0.75 $8$ 3.52e+04 5.61e+04 2.88e+05 1.22e+07 1.78e+09 $16$ 7.29e+05 8.50e+05 4.42e+06 1.64e+08 2.22e+10 $32$ 1.51e+08 1.04e+07 6.92e+07 1.67e+09 1.79e+11 $64$ 3.00e+09 1.17e+08 1.10e+09 1.62e+10 7.08e+13 $128$ 1.39e+10 9.50e+08 1.74e+10 1.16e+11 2.66e+15 t=1 $8$ 3.29e+06 4.49e+04 2.88e+05 1.26e+07 1.10e+09 $16$ 1.02e+07 6.54e+05 4.42e+06 1.08e+08 1.95e+10 $32$ 5.58e+07 9.37e+06 6.92e+07 1.41e+09 1.59e+13 $64$ 3.71e+08 9.56e+07 1.10e+09 1.06e+10 3.93e+13 $128$ 8.04e+11 8.42e+08 1.74e+10 9.95e+10 2.73e+15
 $N_s$ $\mu^+=0.01$ $\mu^+=0.1$ $\mu^+=1$ $\mu^+=10$ $\mu^+=100$ t=0.25 $8$ 4.92e+05 7.56e+04 2.88e+05 1.39e+07 1.32e+09 $16$ 1.43e+06 9.20e+05 4.42e+06 1.12e+08 1.40e+10 $32$ 3.29e+08 6.58e+06 6.92e+07 9.08e+08 4.65e+11 $64$ 7.54e+10 7.37e+07 1.10e+09 8.48e+09 8.48e+12 $128$ 1.43e+12 7.42e+08 1.74e+10 8.16e+10 6.26e+14 t=0.75 $8$ 3.52e+04 5.61e+04 2.88e+05 1.22e+07 1.78e+09 $16$ 7.29e+05 8.50e+05 4.42e+06 1.64e+08 2.22e+10 $32$ 1.51e+08 1.04e+07 6.92e+07 1.67e+09 1.79e+11 $64$ 3.00e+09 1.17e+08 1.10e+09 1.62e+10 7.08e+13 $128$ 1.39e+10 9.50e+08 1.74e+10 1.16e+11 2.66e+15 t=1 $8$ 3.29e+06 4.49e+04 2.88e+05 1.26e+07 1.10e+09 $16$ 1.02e+07 6.54e+05 4.42e+06 1.08e+08 1.95e+10 $32$ 5.58e+07 9.37e+06 6.92e+07 1.41e+09 1.59e+13 $64$ 3.71e+08 9.56e+07 1.10e+09 1.06e+10 3.93e+13 $128$ 8.04e+11 8.42e+08 1.74e+10 9.95e+10 2.73e+15
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