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doi: 10.3934/era.2021032
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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 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, doi: 10.3934/era.2021032
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]

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. DuarteR. 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

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[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. GuoT. 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. GuoT. 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. 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

[16]

X. HeT. 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. HeT. LinY. 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

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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. LiT. 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. LinY. 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. LinY. 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. LinD. 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. 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. 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. LundbergP. 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. 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]

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. DuarteR. 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. GuoT. 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. GuoT. 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. 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

[16]

X. HeT. 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. HeT. LinY. 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. LiT. 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. LinY. 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. LinY. 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. LinD. 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. 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. 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. LundbergP. 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

Figure 1.  The geometrical setup of a moving interface problem
Figure 2.  From left: an interface-fitted mesh and an unfitted mesh
Figure 3.  Types of interface elements. From left: Type Ⅰ, Type Ⅱ, Type Ⅲ
Figure 4.  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
Figure 5.  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
Figure 6.  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} $
Table 1.  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
Table 2.  $ 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
Table 3.  $ 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
Table 4.  $ 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
Table 5.  $ 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
Table 6.  $ 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
Table 7.  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
Table 8.  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
Table 9.  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
Table 10.  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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