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Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces. (English) Zbl 1120.65333
Summary: Elliptic problems with sharp-edged interfaces, thin-layered interfaces and interfaces that intersect with geometric boundary, are notoriously challenging to existing numerical methods, particularly when the solution is highly oscillatory. This work generalizes the matched interface and boundary method previously designed for solving elliptic problems with curved interfaces to the aforementioned problems. We classify these problems into five distinct topological relations involving the interfaces and the Cartesian mesh lines. Flexible strategies are developed to systematically extend the computational domains near the interface so that the standard central finite difference scheme can be applied without the loss of accuracy. Fictitious values on the extended domains are determined by enforcing the physical jump conditions on the interface according to the local topology of the irregular point. The concepts of primary and secondary fictitious values are introduced to deal with sharp-edged interfaces. For corner singularity or tip singularity, an appropriate polynomial is multiplied to the solution to remove the singularity. Extensive numerical experiments confirm the designed second order convergence of the proposed method.

MSC:
65N06Finite difference methods (BVP of PDE)
65N12Stability and convergence of numerical methods (BVP of PDE)
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
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References:
[1] Adams, L.; Li, Z. L.: The immersed interface/multigrid methods for interface problems. SIAM J. Sci. comput. 24, 463-479 (2002) · Zbl 1014.65099
[2] Babuška, I.: The finite element method for elliptic equations with discontinuous coefficients. Computing 5, 207-213 (1970) · Zbl 0199.50603
[3] Babuška, I.; Sauter, S. A.: Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave number?. SIAM J. Numer. anal. 34, 2392-2423 (1997) · Zbl 0894.65050
[4] Baetke, F.; Werner, H.; Wengle, H.: Numerical-simulation of turbulent-flow over surface-mounted obstacles with sharp edges and corners. J. wind engng. Indust. aerodyn. 35, 129-147 (1990)
[5] Bao, G.; Wei, G. W.; Zhao, S.: Numerical solution of the Helmholtz equation with high wave numbers. Int. J. Numer. methods engng. 59, 389-408 (2004) · Zbl 1043.65132
[6] Ben Ameur, H.; Burger, M.; Hackl, B.: Level set methods for geometric inverse problems in linear elasticity. Inverse problems 20, 673-696 (2004) · Zbl 1086.35117
[7] Berthelsen, P. A.: A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions. J. comput. Phys. 197, 364-386 (2004) · Zbl 1052.65100
[8] Biros, G.; Ying, L. X.; Zorin, D.: A fast solver for the Stokes equations with distributed forces in complex geometries. J. comput. Phys. 193, 317-348 (2004) · Zbl 1047.76065
[9] Bramble, J.; King, J.: A finite element method for interface problems in domains with smooth boundaries and interfaces. Adv. comput. Math. 6, 109-138 (1996) · Zbl 0868.65081
[10] Braess, D.: Finite elements: theory, fast solvers, and applications in solid mechanics. (1997) · Zbl 0894.65054
[11] Cai, W.; Deng, S. Z.: An upwinding embedded boundary method for Maxwell’s equations in media with material interfaces: 2D case. J. comput. Phys. 190, 159-183 (2003) · Zbl 1031.78005
[12] Caorsi, S.; Pastorino, M.; Raffetto, M.: Electromagnetic scattering by a conducting strip with a multilayer elliptic dielectric coating. IEEE trans. Electromag. compat. 41, 335-343 (1999)
[13] Cendes, Z. J.; Shenton, D. N.; Shahnasser, H.: Magnetic field computation using delaney triangulation and complementary finite element methods. IEEE trans. Magn. 19, 2251-2554 (1983)
[14] Deng, S. Z.; Ito, K.; Li, Z. L.: Three-dimensional elliptic solvers for interface problems and applications. J. comput. Phys. 184, 215-243 (2003) · Zbl 1016.65072
[15] Dumett, M. A.; Keener, J. P.: An immersed interface method for solving anisotropic elliptic boundary value problems in three dimensions. SIAM J. Sci. comput. 25, 348-367 (2003) · Zbl 1071.65137
[16] Fadlun, E. A.; Verzicco, R.; Orlandi, P.; Mohd-Yusof, J.: Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. J. comput. Phys. 161, 30-60 (2000) · Zbl 0972.76073
[17] Fedkiw, R. P.; Aslam, T.; Merriman, B.; Osher, S.: A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. comput. Phys. 152, 457-492 (1999) · Zbl 0957.76052
[18] Fogelson, A. L.; Keener, J. P.: Immersed interface methods for Neumann and related problems in two and three dimensions. SIAM J. Sci. comput. 22, 1630-1654 (2000) · Zbl 0982.65112
[19] Francois, M.; Shyy, W.: Computations of drop dynamics with the immersed boundary method, part 2: drop impact and heat transfer. Numer. heat trans. Part B --- fund. 44, 119-143 (2003)
[20] Gibou, F.; Fedkiw, R. P.: A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem. J. comput. Phys. 202, 577-601 (2005) · Zbl 1061.65079
[21] Griffith, B. E.; Peskin, C. S.: On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficiently smooth problems. J. comput. Phys. 208, 75-105 (2005) · Zbl 1115.76386
[22] G. Guyomarch, C.-O. Lee, A discontinuous Galerkin method for elliptic interface problems with application to electroporation, AIST DAM Research Report, 04-14, 2004.
[23] Hadley, G. R.: High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces. J. lightw. Technol. 20, 1210-1218 (2002)
[24] Hesthaven, J. S.: High-order accurate methods in time-domain computational electromagnetics. A review. Adv. imaging electron phys. 127, 59-123 (2003)
[25] Hou, S.; Liu, X. -D.: A numerical method for solving variable coefficient elliptic equation with interfaces. J. comput. Phys. 202, 411-445 (2005) · Zbl 1061.65123
[26] Hou, T. Y.; Li, Z. L.; Osher, S.; Zhao, H.: A hybrid method for moving interface problems with application to the Hele -- Shaw flow. J. comput. Phys. 134, 236-252 (1997) · Zbl 0888.76067
[27] Huang, H.; Li, Z. L.: Convergence analysis of the immersed interface method. IMA J. Numer. anal. 19, 583-608 (1999) · Zbl 0940.65114
[28] Hunter, J. K.; Li, Z. L.; Zhao, H.: Reactive autophobic spreading of drops. J. comput. Phys. 183, 335-366 (2002) · Zbl 1046.76036
[29] Iaccarino, G.; Verzicco, R.: Immersed boundary technique for turbulent flow simulations. Appl. mech. Rev. 56, 331-347 (2003)
[30] Jin, S.; Wang, X. L.: Robust numerical simulation of porosity evolution in chemical vapor infiltration II. Two-dimensional anisotropic fronts. J. comput. Phys. 179, 557-577 (2002) · Zbl 1130.76385
[31] Johansen, H.; Colella, P.: A Cartesian grid embedding boundary method for Poisson’s equation on irregular domains. J. comput. Phys. 147, 60-85 (1998) · Zbl 0923.65079
[32] Kandilarov, J. D.: Immersed interface method for a reaction -- diffusion equation with a moving own concentrated source. Lecture notes comput. Sci. 2542, 506-513 (2003) · Zbl 1032.65093
[33] Lai, M. C.; Peskin, C. S.: An immersed boundary method with formal second-order accuracy and reduced numerical viscosity. J. comput. Phys. 160, 705-719 (2000) · Zbl 0954.76066
[34] Lee, L.; Leveque, R. J.: An immersed interface method for incompressible Navier -- Stokes equations. SIAM J. Sci. comput. 25, 832-856 (2003) · Zbl 1163.65322
[35] Leveque, R. J.; Li, Z. L.: The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. anal. 31, 1019-1044 (1994) · Zbl 0811.65083
[36] Li, Z. L.: A fast iterative algorithm for elliptic interface problems. SIAM J. Numer. anal. 35, 230-254 (1998) · Zbl 0915.65121
[37] Li, Z. L.; Ito, K.: Maximum principle preserving schemes for interface problems with discontinuous coefficients. SIAM J. Sci. comput. 23, 339-361 (2001) · Zbl 1001.65115
[38] Li, Z. L.; Lin, T.; Wu, X. H.: New Cartesian grid methods for interface problems using the finite element formulation. Numer. math. 96, 61-98 (2003) · Zbl 1055.65130
[39] Li, Z. L.; Lubkin, S. R.: Numerical analysis of interfacial two-dimensional Stokes flow with discontinuous viscosity and variable surface tension. Int. J. Numer. methods fluid 37, 525-540 (2001) · Zbl 0996.76068
[40] Li, Z. L.; Wang, W. -C.; Chern, I. -L.; Lai, M. -C.: New formulations for interface problems in polar coordinates. SIAM J. Sci. comput. 25, 224-245 (2003) · Zbl 1040.65087
[41] Linnick, M. N.; Fasel, H. F.: A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains. J. comput. Phys. 204, 157-192 (2005) · Zbl 1143.76538
[42] Lombard, B.; Piraux, J.: How to incorporate the spring-mass conditions in finite-difference schemes. SIAM J. Sci. comput. 24, 1379-1407 (2003) · Zbl 1035.35062
[43] Macak, E. B.; Munz, W. D.; Rodenburg, J. M.: Plasma -- surface interaction at sharp edges and corners during ion-assisted physical vapor deposition. Part I: Edge-related effects and their influence on coating morphology and composition. J. appl. Phys. 94, 2829-2836 (2003)
[44] 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
[45] Mckenney, A.; Greengard, L.; Mayo, A.: A fast Poisson solver for complex geometries. J. comput. Phys. 118, 348-355 (1995) · Zbl 0823.65115
[46] Miniowitz, R.; Webb, J. P.: Covariant-projection quadrilateral elements for the analysis of wave-guides with sharp edges. IEEE trans. Microwave theory techn. 39, 501-505 (1991)
[47] Mittal, R.; Iaccarino, G.: Immersed boundary methods. Annu. rev. Fluid mech. 37, 236-261 (2005) · Zbl 1117.76049
[48] Pan, G.; Tong, M.; Gilbert, B.: Multiwavelet based moment method under discrete Sobolev-type norm. Microwave opt. Techn. lett. 40, 47-50 (2004)
[49] Pantic-Tanner, Z.; Savage, J. Z.; Tanner, D. R.; Peterson, A. F.: Two-dimensional singular vector elements for finite-element analysis. IEEE trans. Microwave theory techn. 46, 178-184 (1998)
[50] Peskin, C. S.: Numerical analysis of blood flow in heart. J. comput. Phys. 25, 220-252 (1977) · Zbl 0403.76100
[51] Peskin, C. S.: Lectures on mathematical aspects of physiology. Lect. appl. Math. 19, 69-107 (1981) · Zbl 0461.92004
[52] Peskin, C. S.; Mcqueen, D. M.: A 3-dimensional computational method for blood-flow in the heart. 1. Immersed elastic fibers in a viscous incompressible fluid. J. comput. Phys. 81, 372-405 (1989) · Zbl 0668.76159
[53] Schulz, M.; Steinebach, G.: Two-dimensional modelling of the river rhine. J. comput. Appl. math. 145, 11-20 (2002) · Zbl 1072.86001
[54] Sethian, J. A.; Wiegmann, A.: Structural boundary design via level set and immersed interface methods. J. comput. Phys. 163, 489-528 (2000) · Zbl 0994.74082
[55] Tornberg, A. K.; Engquist, B.: Numerical approximations of singular source terms in differential equations. J. comput. Phys. 200, 462-488 (2004) · Zbl 1115.76392
[56] Van Rens, B. J. E.; Brekelmans, W. A. M.; Baaijens, F. P. T.: Modelling friction near sharp edges using a Eulerian reference frame: application to aluminum extrusion. Int. J. Numer. methods engng. 54, 453-471 (2002) · Zbl 1098.74722
[57] Voorde, J. V.; Vierendeels, J.; Dick, E.: Flow simulations in rotary volumetric pumps and compressors with the fictitious domain method. J. comput. Appl. math. 168, 491-499 (2004) · Zbl 1058.76052
[58] Walther, J. H.; Morgenthal, G.: An immersed interface method for the vortex-in-cell algorithm. J. turbulence 3 (2002) · Zbl 1082.76587
[59] Wiegmann, A.; Bube, K. P.: 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
[60] Zhao, S.; Wei, G. W.: High order FDTD methods via derivative matching for Maxwell’s equations with material interfaces. J. comput. Phys. 200, 60-103 (2004) · Zbl 1050.78018
[61] Zhou, Y. C.; Zhao, S.; Feig, M.; Wei, G. W.: High order matched interface and boundary (MIB) schemes for elliptic equations with discontinuous coefficients and singular sources. J. comput. Phys. 231, 1-30 (2006) · Zbl 1089.65117
[62] Zhou, Y. C.; Wei, G. W.: On the fictitious-domain and interpolation formulations of the matched interface and boundary (MIB) method. J. comput. Phys. 219, 228-246 (2006) · Zbl 1105.65108