A Cartesian grid embedded boundary method for Poisson’s equation on irregular domains. (English) Zbl 0923.65079

The authors present a finite-volume method for the Poisson equation on a curved planar region covered by a regular grid. A special discretization is used on computational cells which extend beyond the domain. It is shown that the method is second-order accurate, and numerical examples suggest that the condition number of the matrix is bounded independent of the fact that the intersection of some computational cells with the domain may be very small.


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation


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[3] Almgren, A. S.; Bell, J. B.; Colella, P.; Howell, L. H.; Welcome, M. L., A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations, J. Comput. Phys., 142, 1 (1998) · Zbl 0933.76055
[5] Almgren, A.; Bell, J. B.; Colella, P.; Marthaler, T., A Cartesian grid projection method for the incompressible Euler equations in complex geometries, SIAM J. Sci. Comput., 18, 1289 (1998) · Zbl 0910.76040
[6] Babuska, I., Modeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations (1995)
[8] Berger, M. J.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82, 64 (1989) · Zbl 0665.76070
[9] Berger, M. J.; Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., 53, 484 (1984) · Zbl 0536.65071
[10] Bramble, J. H.; Ewing, R. E.; Pasciak, J. E.; Shen, J., The analysis of multigrid algorithms for cell-centered finite difference methods, Adv. Comput. Math., 5, 15 (1996) · Zbl 0848.65082
[11] Briggs, W., A Multigrid Tutorial (1987)
[12] Chan, T. F.; Mathew, T. P., Domian decomposition algorithms, Acta Numer., 41 (1994)
[13] Chern, I.; Colella, P., A Convervative Front Tracking Method for Hyperbolic Conservation Laws (1987)
[14] Colella, P.; Henderson, L. F., The von Neumann paradox for the diffraction of weak shock waves, J. Fluid Mech., 213, 71 (1990)
[15] Crutchfield, W. Y.; Welcome, M. L., Object-oriented implementations of adaptive mesh refinement algorithms, Sci. Programming, 2, 145 (1993)
[16] Demkowicz, L.; Oden, J. T.; Rachowicz, W.; Hardy, O., Toward a universal h-p adaptive finite element strategy, Comput. Methods Appl. Mech. Eng., 77, 79 (1989) · Zbl 0723.73074
[17] Greengard, L.; Lee, J., A direct Poisson solver of arbitrary order accuracy, J. Comput. Phys., 125, 415 (1996) · Zbl 0851.65090
[18] Chesshire, G.; Henshaw, W. D., Composite overlapping meshes for the solution of partial differential equations, J. Comput. Phys., 90, 1 (1990) · Zbl 0709.65090
[19] Gilbarg, N.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1977) · Zbl 0361.35003
[20] Howell, L. H.; Bell, J. B., An adaptive-mesh projection method for viscous incompressible flow, SIAM J. Sci. Comp., 18, 996 (1997) · Zbl 0901.76057
[21] Johnson, C., Numerical Solution of Partial Differential Equations by the Finite Element Method (1987)
[22] LeVeque, R. J.; Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31, 1019 (1994) · Zbl 0811.65083
[24] Puckett, E. G.; Almgren, A. S.; Bell, J. B.; Marcus, D. L.; Rider, W. J., A high-order projection method for tracking fluid interfaces in variable density incompressible flows, J. Comput. Phys., 130, 269 (1997) · Zbl 0872.76065
[25] Johansen, H., Cartesian Grid Embedded Boundary Finite Difference Methods for Elliptic and Parabolic Differential Equations on Irregular Domains, (1997)
[26] Li, Z., The Immersed Interface Method—A Numerical Approach for Partial Differential Equations with Interfaces, (1994)
[27] Li, Z., A fast iterative method for elliptic interface problems, SIAM J. Numer. Anal., 35, 230 (1998) · Zbl 0915.65121
[28] Martin, D. F.; Cartwright, K. L., Solving Poisson’s Equation Using Adaptive Mesh Refinement (1996)
[29] Mayo, A., The rapid evaluation of volume integrals of potential theory on general regions, J. Comput. Phys., 100, 236 (1992) · Zbl 0772.65012
[30] McKenney, A.; Greengard, L.; Mayo, A., A fast Poisson solver for complex geometries, J. Comput. Phys., 118, 348 (1995) · Zbl 0823.65115
[31] Minion, M., A projection method for locally refined grids, J. Comput. Phys., 127, 158 (1996) · Zbl 0859.76047
[32] Pember, R. B.; Bell, J. B.; Colella, P.; Crutchfield, W. Y.; Welcome, M., An adaptive Cartesian grid method for unsteady compressible flow in irregular regions, J. Comput. Phys., 120, 278 (1995) · Zbl 0842.76056
[33] Peskin, C. S.; Printz, B. F., Improved volume conservation in the computation of flows with immersed elastic boundaries, J. Comput. Phys., 105, 33 (1993) · Zbl 0762.92011
[35] Schmidt, A., Computation of three dimensional dendrites with finite elements, J. Comput. Phys., 125, 293 (1996) · Zbl 0844.65096
[36] Young, D. P.; Melvin, R. G.; Bieterman, M. B.; Johnson, F. T.; Samant, S. S.; Bussoletti, J. E., A locally refined rectangular grid finite element method: Application to computational fluid dynamics and computational physics, J. Comput. Phys., 62, 1 (1991) · Zbl 0709.76078
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