# zbMATH — the first resource for mathematics

A source term method for Poisson problems on irregular domains. (English) Zbl 1422.65327
Summary: This paper presents a finite difference method for solving Poisson problems on a two-dimensional irregular domain. An implicit, level set representation of the domain boundary is assumed, as well as a Cartesian grid that is not fitted to the domain. The algorithm is based on a scheme for interface problems which captures the jump conditions via singular source terms. This paper adapts that method to deal with boundary value problems by employing a simple iterative process that simultaneously enforces the boundary condition and solves for an unknown jump condition. The benefit and novelty of this method is that the boundary condition is captured via easily implemented source terms. The system of equations that results at each iteration can be solved using a FFT-based fast Poisson solver. The scheme can accommodate Dirichlet, Neumann, and Robin boundary conditions. We first address the constant coefficient Poisson equation, and then extend the scheme to accommodate the variable coefficient equation. Numerical examples indicate second order accuracy (or close to it) for the solution. The method also produces useful gradient approximations, but with generally lower convergence rates.

##### MSC:
 65N06 Finite difference methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Full Text:
##### References:
 [1] Askham, T.; Cerfon, A., An adaptive fast multipole accelerated Poisson solver for complex geometries, J. Comput. Phys., 344, 1-22, (2017) · Zbl 1380.65413 [2] Aslam, T., A partial differential equation approach to multidimensional extrapolation, J. Comput. Phys., 193, 349-355, (2003) · Zbl 1036.65002 [3] Aslam, T.; Luo, S.; Zhao, H., A static PDE approach to multi-dimensional extrapolations using fast sweeping methods, SIAM J. Sci. Comput., 36, A2907-A2928, (2014) · Zbl 1315.65093 [4] Beale, J.; Lai, M., A method for computing nearly singular integrals, SIAM J. Numer. Anal., 38, 1902-1925, (2001) · Zbl 0988.65025 [5] Beale, J.; Layton, A., On the accuracy of finite difference methods for elliptic problems with interfaces, Commun. Appl. Math. Comput. Sci., 1, 91-119, (2006) · Zbl 1153.35319 [6] Bedrosian, J.; von Brecht, J. J.; Zhu, S.; Sifakis, E.; Teran, J., A second order virtual node method for elliptic problems with interfaces and irregular domains, J. Comput. Phys., 229, 6405-6426, (2010) · Zbl 1197.65168 [7] Calhoun, D., A Cartesian grid method for solving the streamfunction-vorticity equation in irregular regions, J. Comput. Phys., 176, 231-275, (2002) · Zbl 1130.76371 [8] Chopp, D., Some improvements of the fast marching method, SIAM J. Sci. Comput., 23, 230-244, (2001) · Zbl 0991.65105 [9] Buzbee, B.; Dorr, F.; George, J.; Golub, G., The direct solution of the discrete Poisson equation on irregular domains, SIAM J. Numer. Anal., 8, 722-736, (1971) · Zbl 0231.65083 [10] Gibou, F.; Fedkiw, R., A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to Stefan problems, J. Comput. Phys., 202, 577-601, (2002) · Zbl 1061.65079 [11] Gibou, F.; Fedkiw, R.; Cheng, K.; Kang, M., A second order accurate symmetric discretization of the Poisson equation on irregular domains, J. Comput. Phys., 176, 1-23, (2003) [12] Gibou, F.; Min, C.; Fedkiw, R., High resolution sharp computational methods for elliptic and parabolic problems in complex geometries, J. Sci. Comput., 54, 369-413, (2013) · Zbl 1263.65093 [13] Guittet, A.; Lepilliez, M.; Tanguy, S.; Gibou, F., Solving elliptic problems with discontinuities on irregular domains — the Voronoi interface method, J. Comput. Phys., 298, 747-765, (2015) · Zbl 1349.65579 [14] Ito, K.; Kunisch, K.; Li, Z., Level-set function approach to an inverse interface problem, Inverse Probl., 17, 1225-1242, (2001) · Zbl 0986.35130 [15] Johansen, H.; Colella, P., A cartesian grid embedded boundary method for Poisson’s equation on irregular domains, J. Comput. Phys., 147, 60-85, (1998) · Zbl 0923.65079 [16] Jomaa, Z.; Macaskill, C., The embedded finite difference method for the Poisson equation in a domain with an irregular boundary and Dirichlet boundary conditions, J. Comput. Phys., 202, 488-506, (2005) · Zbl 1061.65107 [17] Jomaa, Z.; Macaskill, C., Numerical solution of the 2d Poisson equation on an irregular domain with Robin boundary conditions, ANZIAM J., 50, C413-C428, (2008) · Zbl 1359.65290 [18] Kublik, C.; Tanushev, N. M.; Tsai, R., An implicit interface boundary integral method for Poisson’s equation on arbitrary domains, J. Comput. Phys., 247, 269-311, (2013) · Zbl 1349.65661 [19] Leveque, R.; Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31, 1019-1044, (1994) · Zbl 0811.65083 [20] LeVeque, R.; Li, Z., Immersed interface methods for Stokes flow with elastic boundaries or surface tension, SIAM J. Sci. Comput., 18, 709-735, (1997) · Zbl 0879.76061 [21] Li, Z., A fast iterative algorithm for elliptic interface problems, SIAM J. Numer. Anal., 35, 230-254, (1998) · Zbl 0915.65121 [22] Li, Z.; Ito, K., The immersed interface method, Frontiers in Applied Mathematics, (2006), SIAM Philadelphia, PA [23] Li, Z.; Wang, W., A fast finite difference method for solving Navier-Stokes equations on irregular domains, Commun. Math. Sci., 1, 180-196, (2003) · Zbl 1082.76077 [24] Li, Z.; Zhao, H.; Gao, H., A numerical study of electro-migration voiding by evolving level set functions on a fixed Cartesian grid, J. Comput. Phys., 152, 281-304, (1999) · Zbl 0956.78016 [25] Liu, X.; Fedkiw, R.; Kang, M., A boundary condition capturing method for Poisson’s equation on irregular domains, J. Comput. Phys., 160, 151-178, (2000) · Zbl 0958.65105 [26] Marques, A.; Nave, J.; Rosales, R., A correction function method for Poisson problems with interface jump conditions, J. Comput. Phys., 230, 7567-7597, (2011) · Zbl 1453.35054 [27] Mayo, A., The fast solution of Poisson’s and the biharmonic equations on irregular regions, SIAM J. Numer. Anal., 21, 285-299, (1984) · Zbl 1131.65303 [28] McKenney, A.; Greengard, L.; Mayo, A., A fast Poisson solver for complex geometries, J. Comput. Phys., 118, 348-355, (1995) · Zbl 0823.65115 [29] Min, C.; Gibou, F., A second order accurate level set method on non-graded adaptive Cartesian grids, J. Comput. Phys., 225, 300-321, (2007) · Zbl 1122.65077 [30] Osher, S.; Fedkiw, R., Level set methods and dynamic implicit surfaces, (2003), Springer-Verlag New York, NY · Zbl 1026.76001 [31] Osher, S.; Sethian, J., Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79, 12-49, (1988) · Zbl 0659.65132 [32] Papac, J.; Gibou, F.; Ratsch, C., Efficient symmetric discretization for the Poisson, heat and Stefan-type problems with Robin boundary conditions, J. Comput. Phys., 229, 875-889, (2010) · Zbl 1182.65140 [33] Russell, D.; Wang, Z., A Cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow, J. Comput. Phys., 191, 177-205, (2003) · Zbl 1160.76389 [34] Sethian, J., Level set methods and fast marching methods, (1999), Cambridge University Press Cambridge · Zbl 0929.65066 [35] Stein, D.; Guy, R.; Thomases, B., Immersed boundary smooth extension (IBSE): a high-order method for solving incompressible flows in arbitrary smooth domains, J. Comput. Phys., 335, 155-178, (2017) · Zbl 1375.76038 [36] Tornberg, A.; Engquist, B., Numerical approximations of singular source terms in differential equations, J. Comput. Phys., 200, 462-488, (2004) · Zbl 1115.76392 [37] Towers, J., Two methods for discretizing a delta function supported on a level set, J. Comput. Phys., 220, 915-931, (2007) · Zbl 1115.65028 [38] Towers, J., Finite difference methods for approximating heaviside functions, J. Comput. Phys., 228, 3478-3489, (2009) · Zbl 1171.65014 [39] Towers, J., Finite difference methods for discretizing singular source terms in a Poisson interface problem, Contemp. Math., 526, 359-389, (2010) · Zbl 1216.65143 [40] Wiegmann, A.; Bube, K., The explicit-jump immersed interface method: finite difference methods for PDEs with piecewise smooth solutions, SIAM J. Numer. Anal., 37, 827-862, (2000) · Zbl 0948.65107
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.