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A second-order boundary condition capturing method for solving the elliptic interface problems on irregular domains. (English) Zbl 1427.65327

Summary: A second-order boundary condition capturing method is presented for the elliptic interface problem with jump conditions in the solution and its normal derivative. The proposed method is an extension of the work in [X.-D. Liu et al., J. Comput. Phys. 160, No. 1, 151–178 (2000; Zbl 0958.65105)] to a higher order. The motivation of proposed method is that the approximated value at the interface can be reconstructed by proper interpolation based on the level set representation from F. Gibou et al. [J. Comput. Phys. 176, No. 1, 205–227 (2002; Zbl 0996.65108)]. A second-order accurate method is constructed, both in the solution and its gradient, using second-order finite difference approximation. Several numerical results demonstrate that the proposed method is indeed second-order accurate in the solution and its gradient in the \(L^2\) and \(L^{\infty}\) norms.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
35J15 Second-order elliptic equations
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[1] Chen, X., Feng, X., Li, Z.: A direct method for accurate solution and gradient computations for elliptic interface problems. Numer. Algorithms 80(3), 709-740 (2019) · Zbl 1412.65179
[2] Chern, I.L., Shu, Y.C.: A coupling interface method for elliptic interface problems. J. Comput. Phys. 225(2), 2138-2174 (2007) · Zbl 1123.65108
[3] Coco, A., Russo, G.: Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface. J. Comput. Phys. 361, 299-330 (2018) · Zbl 1422.65306
[4] Gibou, F., Fedkiw, R.P., Cheng, L.T., Kang, M.: A second-order-accurate symmetric discretization of the Poisson equation on irregular domains. J. Comput. Phys. 176(1), 205-227 (2002) · Zbl 0996.65108
[5] 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
[6] Lee, L., LeVeque, R.J.: An immersed interface method for incompressible Navier-Stokes equations. SIAM J. Sci. Comput. 25(3), 832-856 (2003) · Zbl 1163.65322
[7] Leveque, R.J., Li, Z.: The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal. 31(4), 1019-1044 (1994) · Zbl 0811.65083
[8] LeVeque, R.J., Li, Z.: Immersed interface methods for Stokes flow with elastic boundaries or surface tension. SIAM J. Sci. Comput. 18(3), 709-735 (1997) · Zbl 0879.76061
[9] Li, Z.: A fast iterative algorithm for elliptic interface problems. SIAM J. Numer. Anal. 35(1), 230-254 (1998) · Zbl 0915.65121
[10] Li, Z., Lai, M.C.: The immersed interface method for the Navier-Stokes equations with singular forces. J. Comput. Phys. 171(2), 822-842 (2001) · Zbl 1065.76568
[11] Li, Z., Ji, H., Chen, X.: Accurate solution and gradient computation for elliptic interface problems with variable coefficients. SIAM J. Numer. Anal. 55(2), 570-597 (2017). https://doi.org/10.1137/15M1040244 · Zbl 1362.76037 · doi:10.1137/15M1040244
[12] Linnick, M.N., Fasel, H.F.: A high-order immersed interface method for simulating unsteady incompressible flows on irregular domains. J. Comput. Phys. 204(1), 157-192 (2005) · Zbl 1143.76538
[13] Liu, X.D., Fedkiw, R.P., Kang, M.: A boundary condition capturing method for Poisson’s equation on irregular domains. J. Comput. Phys. 160(1), 151-178 (2000) · Zbl 0958.65105
[14] Marques, A.N., Nave, J.C., Rosales, R.R.: A correction function method for Poisson problems with interface jump conditions. J. Comput. Phys. 230(20), 7567-7597 (2011) · Zbl 1453.35054
[15] Marques, A.N., Nave, J.C., Rosales, R.R.: High order solution of Poisson problems with piecewise constant coefficients and interface jumps. J. Comput. Phys. 335, 497-515 (2017) · Zbl 1380.65224
[16] Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12-49 (1988) · Zbl 0659.65132
[17] Saad, Y.: Iterative Methods for Sparse Linear Systems, vol. 82. SIAM, Philadelphia (2003) · Zbl 1031.65046
[18] Seo, J., Ha, Sy, Min, C.: Convergence analysis in the maximum norm of the numerical gradient of the Shortley-Weller method. J. Sci. Comput. 74(2), 631-639 (2018) · Zbl 1397.65227
[19] Shortley, G.H., Weller, R.: The numerical solution of Laplace’s equation. J. Appl. Phys. 9(5), 334-348 (1938) · Zbl 0019.03801
[20] 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(3), 827-862 (2000) · Zbl 0948.65107
[21] Yoon, G., Min, C.: A review of the supra-convergences of Shortley-Weller method for Poisson equation. J. KSIAM 18(1), 51-60 (2014) · Zbl 1317.65219
[22] Yoon, G., Min, C.: Analyses on the finite difference method by Gibou et al. for Poisson equation. J. Comput. Phys. 280, 184-194 (2015) · Zbl 1349.65573
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