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doi: 10.3934/era.2021043
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Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions

1. 

Department of Mathematics, Ajou University, Suwon 16499, Korea

2. 

Department of Mathematics, Myongji University, Yongin 17058, Korea

* Corresponding author

Received  September 2020 Revised  April 2021 Early access June 2021

Fund Project: Jeon was supported by NRF 2018R1D1A1A09082082 and Shin was supported by 2020 Research Fund of Myongji University

We propose an immersed hybrid difference method for elliptic boundary value problems by artificial interface conditions. The artificial interface condition is derived by imposing the given boundary condition weakly with the penalty parameter as in the Nitsche trick and it maintains ellipticity. Then, the derived interface problems can be solved by the hybrid difference approach together with a proper virtual to real transformation. Therefore, the boundary value problems can be solved on a fixed mesh independently of geometric shapes of boundaries. Numerical tests on several types of boundary interfaces are presented to demonstrate efficiency of the suggested method.

Citation: Youngmok Jeon, Dongwook Shin. Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions. Electronic Research Archive, doi: 10.3934/era.2021043
References:
[1]

T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for second-order elliptic problems, Math. Comp., 64 (1995), 943-972.  doi: 10.2307/2153478.  Google Scholar

[2]

D. N. ArnoldF. BrezziB. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), 1749-1779.  doi: 10.1137/S0036142901384162.  Google Scholar

[3]

F. Bassi and S. Rebay, High-order accurate discontinuous finite element solution of the 2D Euler equations, J. Comput. Phys., 138 (1997), 251-285.  doi: 10.1006/jcph.1997.5454.  Google Scholar

[4]

D. Braess, Finite Elements, Theory, Fast Solvers, and Applications in Solid Mechanics, 2$^{nd}$ edition, Cambridge University Press, 2001.  Google Scholar

[5]

S. C. BrennerM. Neilan and L.-Y. Sung, Isoparametric $C^0$ interior penalty methods for plate bending problems on smooth domains, Calcolo, 50 (2013), 35-67.  doi: 10.1007/s10092-012-0057-1.  Google Scholar

[6]

F. BrezziJ. Douglas and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), 217-235.  doi: 10.1007/BF01389710.  Google Scholar

[7]

F. BrezziL. D. Marini and E. Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Math. Models Methods Appl. Sci., 14 (2004), 1893-1903.  doi: 10.1142/S0218202504003866.  Google Scholar

[8]

B. CampT. LinY. Lin and W. Sun, Quadratic immersed finite element spaces and their approximation capabilities, Adv. Comput. Math., 24 (2006), 81-112.  doi: 10.1007/s10444-004-4139-8.  Google Scholar

[9]

B. CockburnD. A. Di Pietro and A. Ern, Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods, ESAIM Math. Model. Numer. Anal., 50 (2016), 635-650.  doi: 10.1051/m2an/2015051.  Google Scholar

[10]

A. Ern and J.-L. Guermond, Discontinuous Galerkin Methods for Friedrichs' systems. I. General theory, SIAM J. Numer. Anal., 44 (2006), 753-778.  doi: 10.1137/050624133.  Google Scholar

[11]

R. P. FedkiwT. AslamB. Merriman and S. Osher, A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys., 152 (1999), 457-492.  doi: 10.1006/jcph.1999.6236.  Google Scholar

[12]

R. P. FedkiwT. Aslam and S. Xu, The ghost fluid method for deflagration and detonation discontinuities, J. Comput. Phys., 154 (1999), 393-427.  doi: 10.1006/jcph.1999.6320.  Google Scholar

[13]

F. FryklundE. Lehto and A.-K. Tornberg, Partition of unity extension of functions on complex domains, J. Comput. Phys., 375 (2018), 57-79.  doi: 10.1016/j.jcp.2018.08.012.  Google Scholar

[14]

Y. Jeon, An immersed hybrid difference method for the elliptic interface equation, preprint. doi: 10.13140/RG.2.2.27746.58566.  Google Scholar

[15]

Y. Jeon, Hybrid difference methods for PDEs, J. Sci. Comput., 64 (2015), 508-521.  doi: 10.1007/s10915-014-9941-y.  Google Scholar

[16]

Y. Jeon, Hybrid spectral difference methods for elliptic equations on exterior domains with the discrete radial absorbing boundary condition, J. Sci. Comput., 75 (2018), 889-905.  doi: 10.1007/s10915-017-0570-0.  Google Scholar

[17]

Y. JeonE.-J. Park and D.-W. Shin, Hybrid spectral difference methods for an elliptic equation, Comput. Methods Appl. Math., 17 (2017), 253-267.  doi: 10.1515/cmam-2016-0043.  Google Scholar

[18]

Y. Jeon and D. Sheen, Upwind hybrid spectral difference methods for steady-state Navier–Stokes equations, in Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, Springer International Publishing (eds. J. Dick, F.Y. Kuo and H. Woźniakowski), (2018), 621–644.  Google Scholar

[19]

L. Krivodonova and M. Berger, High-order accurate implementation of solid wall boundary conditions in curved geometries, J. Comput. Phys., 211 (2006), 492-512.  doi: 10.1016/j.jcp.2005.05.029.  Google Scholar

[20]

S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103 (1992), 16-42.  doi: 10.1016/0021-9991(92)90324-R.  Google Scholar

[21]

R. J. LeVeque and Z. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31 (1994), 1019-1044.  doi: 10.1137/0731054.  Google Scholar

[22]

R. J. LeVeque and Z. Li, Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J. Sci. Comput., 18 (1997), 709-735.  doi: 10.1137/S1064827595282532.  Google Scholar

[23]

A. N. MarquesJ.-C. Nave and R. R. Rosales, High order solution of Poisson problems with piecewise constant coefficients and interface jumps, J. Comput. Phys., 335 (2017), 497-515.  doi: 10.1016/j.jcp.2017.01.029.  Google Scholar

[24]

A. Mayo, The fast solution of Poisson's and the biharmonic equations on irregular regions, SIAM J. Numer. Anal., 21 (1984), 285-299.  doi: 10.1137/0721021.  Google Scholar

[25]

A. Mayo, Fast high order accurate solution of Laplace's equation on irregular regions, SIAM J. Sci. Statist. Comput., 6 (1985), 144-157.  doi: 10.1137/0906012.  Google Scholar

[26] W. Mclean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.   Google Scholar
[27]

C. S. Peskin, The immersed boundary method, Acta Numer., 11 (2002), 479-517.  doi: 10.1017/S0962492902000077.  Google Scholar

[28]

D. Shirokoff and J.-C. Nave, A sharp-interface active penalty method for the incompressible Navier–Stokes equations, J. Sci. Comput., 62 (2015), 53-77.  doi: 10.1007/s10915-014-9849-6.  Google Scholar

[29]

D. B. SteinR. D. Guy and B. Thomases, Immersed boundary smooth extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods, J. Comput. Phys., 304 (2016), 252-274.  doi: 10.1016/j.jcp.2015.10.023.  Google Scholar

[30]

Y. Wang and J. Zhang, Sixth order compact scheme combined with multigrid method and extrapolation technique for 2D Poisson equation, J. Comput. Phys., 228 (2009), 137-146.  doi: 10.1016/j.jcp.2008.09.002.  Google Scholar

[31]

A. Wiegmann and K. P. Bube, The explicit-jump immersed interface method: Finite difference methods for PDEs with piecewise smooth solutions, SIAM J. Numer. Anal., 37 (2000), 827-862.  doi: 10.1137/S0036142997328664.  Google Scholar

[32]

Y. Xie and W. Ying, A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions, J. Comput. Phys., 415 (2020), 109526, 29 pp. doi: 10.1016/j.jcp.2020.109526.  Google Scholar

[33]

W. Ying and C. S. Henriquez, A kernel-free boundary integral method for elliptic boundary value problems, J. Comput. Phys., 227 (2007), 1046-1074.  doi: 10.1016/j.jcp.2007.08.021.  Google Scholar

[34]

W. Ying and W.-C. Wang, A kernel-free boundary integral method for implicitly defined surfaces, J. Comput. Phys., 252 (2013), 606-624.  doi: 10.1016/j.jcp.2013.06.019.  Google Scholar

show all references

References:
[1]

T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for second-order elliptic problems, Math. Comp., 64 (1995), 943-972.  doi: 10.2307/2153478.  Google Scholar

[2]

D. N. ArnoldF. BrezziB. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), 1749-1779.  doi: 10.1137/S0036142901384162.  Google Scholar

[3]

F. Bassi and S. Rebay, High-order accurate discontinuous finite element solution of the 2D Euler equations, J. Comput. Phys., 138 (1997), 251-285.  doi: 10.1006/jcph.1997.5454.  Google Scholar

[4]

D. Braess, Finite Elements, Theory, Fast Solvers, and Applications in Solid Mechanics, 2$^{nd}$ edition, Cambridge University Press, 2001.  Google Scholar

[5]

S. C. BrennerM. Neilan and L.-Y. Sung, Isoparametric $C^0$ interior penalty methods for plate bending problems on smooth domains, Calcolo, 50 (2013), 35-67.  doi: 10.1007/s10092-012-0057-1.  Google Scholar

[6]

F. BrezziJ. Douglas and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), 217-235.  doi: 10.1007/BF01389710.  Google Scholar

[7]

F. BrezziL. D. Marini and E. Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Math. Models Methods Appl. Sci., 14 (2004), 1893-1903.  doi: 10.1142/S0218202504003866.  Google Scholar

[8]

B. CampT. LinY. Lin and W. Sun, Quadratic immersed finite element spaces and their approximation capabilities, Adv. Comput. Math., 24 (2006), 81-112.  doi: 10.1007/s10444-004-4139-8.  Google Scholar

[9]

B. CockburnD. A. Di Pietro and A. Ern, Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods, ESAIM Math. Model. Numer. Anal., 50 (2016), 635-650.  doi: 10.1051/m2an/2015051.  Google Scholar

[10]

A. Ern and J.-L. Guermond, Discontinuous Galerkin Methods for Friedrichs' systems. I. General theory, SIAM J. Numer. Anal., 44 (2006), 753-778.  doi: 10.1137/050624133.  Google Scholar

[11]

R. P. FedkiwT. AslamB. Merriman and S. Osher, A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. Comput. Phys., 152 (1999), 457-492.  doi: 10.1006/jcph.1999.6236.  Google Scholar

[12]

R. P. FedkiwT. Aslam and S. Xu, The ghost fluid method for deflagration and detonation discontinuities, J. Comput. Phys., 154 (1999), 393-427.  doi: 10.1006/jcph.1999.6320.  Google Scholar

[13]

F. FryklundE. Lehto and A.-K. Tornberg, Partition of unity extension of functions on complex domains, J. Comput. Phys., 375 (2018), 57-79.  doi: 10.1016/j.jcp.2018.08.012.  Google Scholar

[14]

Y. Jeon, An immersed hybrid difference method for the elliptic interface equation, preprint. doi: 10.13140/RG.2.2.27746.58566.  Google Scholar

[15]

Y. Jeon, Hybrid difference methods for PDEs, J. Sci. Comput., 64 (2015), 508-521.  doi: 10.1007/s10915-014-9941-y.  Google Scholar

[16]

Y. Jeon, Hybrid spectral difference methods for elliptic equations on exterior domains with the discrete radial absorbing boundary condition, J. Sci. Comput., 75 (2018), 889-905.  doi: 10.1007/s10915-017-0570-0.  Google Scholar

[17]

Y. JeonE.-J. Park and D.-W. Shin, Hybrid spectral difference methods for an elliptic equation, Comput. Methods Appl. Math., 17 (2017), 253-267.  doi: 10.1515/cmam-2016-0043.  Google Scholar

[18]

Y. Jeon and D. Sheen, Upwind hybrid spectral difference methods for steady-state Navier–Stokes equations, in Contemporary Computational Mathematics - A Celebration of the 80th Birthday of Ian Sloan, Springer International Publishing (eds. J. Dick, F.Y. Kuo and H. Woźniakowski), (2018), 621–644.  Google Scholar

[19]

L. Krivodonova and M. Berger, High-order accurate implementation of solid wall boundary conditions in curved geometries, J. Comput. Phys., 211 (2006), 492-512.  doi: 10.1016/j.jcp.2005.05.029.  Google Scholar

[20]

S. K. Lele, Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103 (1992), 16-42.  doi: 10.1016/0021-9991(92)90324-R.  Google Scholar

[21]

R. J. LeVeque and Z. Li, The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31 (1994), 1019-1044.  doi: 10.1137/0731054.  Google Scholar

[22]

R. J. LeVeque and Z. Li, Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J. Sci. Comput., 18 (1997), 709-735.  doi: 10.1137/S1064827595282532.  Google Scholar

[23]

A. N. MarquesJ.-C. Nave and R. R. Rosales, High order solution of Poisson problems with piecewise constant coefficients and interface jumps, J. Comput. Phys., 335 (2017), 497-515.  doi: 10.1016/j.jcp.2017.01.029.  Google Scholar

[24]

A. Mayo, The fast solution of Poisson's and the biharmonic equations on irregular regions, SIAM J. Numer. Anal., 21 (1984), 285-299.  doi: 10.1137/0721021.  Google Scholar

[25]

A. Mayo, Fast high order accurate solution of Laplace's equation on irregular regions, SIAM J. Sci. Statist. Comput., 6 (1985), 144-157.  doi: 10.1137/0906012.  Google Scholar

[26] W. Mclean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.   Google Scholar
[27]

C. S. Peskin, The immersed boundary method, Acta Numer., 11 (2002), 479-517.  doi: 10.1017/S0962492902000077.  Google Scholar

[28]

D. Shirokoff and J.-C. Nave, A sharp-interface active penalty method for the incompressible Navier–Stokes equations, J. Sci. Comput., 62 (2015), 53-77.  doi: 10.1007/s10915-014-9849-6.  Google Scholar

[29]

D. B. SteinR. D. Guy and B. Thomases, Immersed boundary smooth extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods, J. Comput. Phys., 304 (2016), 252-274.  doi: 10.1016/j.jcp.2015.10.023.  Google Scholar

[30]

Y. Wang and J. Zhang, Sixth order compact scheme combined with multigrid method and extrapolation technique for 2D Poisson equation, J. Comput. Phys., 228 (2009), 137-146.  doi: 10.1016/j.jcp.2008.09.002.  Google Scholar

[31]

A. Wiegmann and K. P. Bube, The explicit-jump immersed interface method: Finite difference methods for PDEs with piecewise smooth solutions, SIAM J. Numer. Anal., 37 (2000), 827-862.  doi: 10.1137/S0036142997328664.  Google Scholar

[32]

Y. Xie and W. Ying, A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions, J. Comput. Phys., 415 (2020), 109526, 29 pp. doi: 10.1016/j.jcp.2020.109526.  Google Scholar

[33]

W. Ying and C. S. Henriquez, A kernel-free boundary integral method for elliptic boundary value problems, J. Comput. Phys., 227 (2007), 1046-1074.  doi: 10.1016/j.jcp.2007.08.021.  Google Scholar

[34]

W. Ying and W.-C. Wang, A kernel-free boundary integral method for implicitly defined surfaces, J. Comput. Phys., 252 (2013), 606-624.  doi: 10.1016/j.jcp.2013.06.019.  Google Scholar

Figure 1.  Configuration of the problem domain
cell-point, intercell point">Figure 2.  One dimensional cell configuration: cell-point, intercell point
Figure 3.  An illustration of the extended solutions $ \widetilde{ u}^{\pm}(x) $ in the interface cell $ I_{k-1} $: the solid lines are the original $ u^{\pm}(x) $ and the dotted lines are extensions
Figure 4.  The $ P_3 $-interface cell configuration. The cell interior points are Gaussian points for the interval $ [\eta_{4k}, \eta_{4k+3}] $
Figure 5.  The $ Q_2^* $-grid: $ |R_1| = h_1 \times k_1 $, $ |R_2| = h_2 \times k_1 $, $ |R_3| = h_1 \times k_2 $
Figure 6.  The coherent $ Q_2^* $-grid (left) and $ Q_3^* $-grid cells (center), and the incoherent interface cell (right). There is no rule for the choice of points $ \tau_i $, and a quasi-uniformly distributed set of points along the interface will be reasonable
Figure 7.  Numerical experiments for the 1-d Dirichlet condition on the $ P_3 $-grid. The first two figures represent error and convergence rate, respectively, and the right represents the scaled condition number ($ (\text{Condition Number}) \times h^3 $)
Figure 8.  Numerical results for the 1-d Neumann condition on the $ P_3 $-grid. The scaled condition numbers are almost coincident
Figure 9.  The three different problem domains with the $ Q_3^* $-grid. Grid 1 (left, circle), Grid 2 (middle, rotated ellipse) and Grid 3 (right, five-leafed flower)
Figure 10.  Coherence ratio for the uniform $ Q_3^* $-grids with the circular interface(red), elliptic interface (blue) and flower-like interface (black). The horizontal axis represents the number of subdivisions for each axis of the computational domai
Figure 11.  The $ L_2 $ errors and convergence rates for the Dirichlet condition on the circle of radius $ 0.3 $ with the uniform $ Q_3^* $-grid(Grid 1)
Figure 12.  The $ L_2 $ errors and convergence rates for the Neumann condition on the circle of radius $ 0.3 $ with the uniform $ Q_3^* $-grid(Grid 1)
Figure 13.  The $ L_2 $ errors and convergence rates for the Dirichlet condition on the rotated ellipse with the uniform $ Q_3^* $-grid(Grid 2)
Figure 14.  The $ L_2 $ errors and convergence rates for the Dirichlet condition on the five-leafed flower with the quasi-uniform $ Q_3^* $-grid(Grid 3)
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