Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions

Youngmok Jeon; Dongwook Shin; Youngmok Jeon; Dongwook Shin

doi:10.3934/era.2021043

Electronic Research Archive

2021, Volume 29, Issue 5: 3361-3382. doi: 10.3934/era.2021043

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Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions

Youngmok Jeon ^{1
,
,},
Dongwook Shin ^{2
,}

1.
Department of Mathematics, Ajou University, Suwon 16499, Korea
2.
Department of Mathematics, Myongji University, Yongin 17058, Korea

Received: 01 September 2020 Revised: 01 April 2021 Published: 24 June 2021
Primary: 65L12, 65N06, 65N50

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.
- Artificial interface condition,
- elliptic boundary value problem,
- immersed hybrid difference,
- VR transformation
Citation: Youngmok Jeon, Dongwook Shin. Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions[J]. Electronic Research Archive, 2021, 29(5): 3361-3382. doi: 10.3934/era.2021043

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Abstract

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.

References

[1]	On the implementation of mixed methods as nonconforming methods for second-order elliptic problems. Math. Comp. (1995) 64: 943-972.
[2]	Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. (2002) 39: 1749-1779.
[3]	High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comput. Phys. (1997) 138: 251-285.
[4]	D. Braess, Finite Elements, Theory, Fast Solvers, and Applications in Solid Mechanics, 2$^{nd}$ edition, Cambridge University Press, 2001.
[5]	Isoparametric $C^0$ interior penalty methods for plate bending problems on smooth domains. Calcolo (2013) 50: 35-67.
[6]	Two families of mixed finite elements for second order elliptic problems. Numer. Math. (1985) 47: 217-235.
[7]	Discontinuous Galerkin methods for first-order hyperbolic problems. Math. Models Methods Appl. Sci. (2004) 14: 1893-1903.
[8]	Quadratic immersed finite element spaces and their approximation capabilities. Adv. Comput. Math. (2006) 24: 81-112.
[9]	Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods. ESAIM Math. Model. Numer. Anal. (2016) 50: 635-650.
[10]	Discontinuous Galerkin Methods for Friedrichs' systems. I. General theory. SIAM J. Numer. Anal. (2006) 44: 753-778.
[11]	A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. (1999) 152: 457-492.
[12]	The ghost fluid method for deflagration and detonation discontinuities. J. Comput. Phys. (1999) 154: 393-427.
[13]	Partition of unity extension of functions on complex domains. J. Comput. Phys. (2018) 375: 57-79.
[14]	Y. Jeon, An immersed hybrid difference method for the elliptic interface equation, preprint. doi: 10.13140/RG.2.2.27746.58566
[15]	Hybrid difference methods for PDEs. J. Sci. Comput. (2015) 64: 508-521.
[16]	Hybrid spectral difference methods for elliptic equations on exterior domains with the discrete radial absorbing boundary condition. J. Sci. Comput. (2018) 75: 889-905.
[17]	Hybrid spectral difference methods for an elliptic equation. Comput. Methods Appl. Math. (2017) 17: 253-267.
[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.
[19]	High-order accurate implementation of solid wall boundary conditions in curved geometries. J. Comput. Phys. (2006) 211: 492-512.
[20]	Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. (1992) 103: 16-42.
[21]	The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. (1994) 31: 1019-1044.
[22]	Immersed interface methods for Stokes flow with elastic boundaries or surface tension. SIAM J. Sci. Comput. (1997) 18: 709-735.
[23]	High order solution of Poisson problems with piecewise constant coefficients and interface jumps. J. Comput. Phys. (2017) 335: 497-515.
[24]	The fast solution of Poisson's and the biharmonic equations on irregular regions. SIAM J. Numer. Anal. (1984) 21: 285-299.
[25]	Fast high order accurate solution of Laplace's equation on irregular regions. SIAM J. Sci. Statist. Comput. (1985) 6: 144-157.
[26]	(2000) Strongly Elliptic Systems and Boundary Integral Equations.Cambridge University Press.
[27]	The immersed boundary method. Acta Numer. (2002) 11: 479-517.
[28]	A sharp-interface active penalty method for the incompressible Navier–Stokes equations. J. Sci. Comput. (2015) 62: 53-77.
[29]	Immersed boundary smooth extension: A high-order method for solving PDE on arbitrary smooth domains using Fourier spectral methods. J. Comput. Phys. (2016) 304: 252-274.
[30]	Sixth order compact scheme combined with multigrid method and extrapolation technique for 2D Poisson equation. J. Comput. Phys. (2009) 228: 137-146.
[31]	The explicit-jump immersed interface method: Finite difference methods for PDEs with piecewise smooth solutions. SIAM J. Numer. Anal. (2000) 37: 827-862.
[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
[33]	A kernel-free boundary integral method for elliptic boundary value problems. J. Comput. Phys. (2007) 227: 1046-1074.
[34]	A kernel-free boundary integral method for implicitly defined surfaces. J. Comput. Phys. (2013) 252: 606-624.

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