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

  • * Corresponding author

    * Corresponding author

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

Abstract / Introduction Full Text(HTML) Figure(6) / Table(10) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 65N15, 65N30; Secondary: 35R05.

    Citation:

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  • 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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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