A Semi-Uniform Multigrid Algorithm for Solving Elliptic Interface Problems

References

[1] L. Adams and T. P. Chartier, New geometric immersed interface multigrid solvers, SIAM J. Sci. Comput. 25 (2004), no. 5, 1516–1533. 10.1137/S1064827503421707Search in Google Scholar

[2] L. Adams and Z. Li, The immersed interface/multigrid methods for interface problems, SIAM J. Sci. Comput. 24 (2002), no. 2, 463–479. 10.1137/S1064827501389849Search in Google Scholar

[3] I. Babuška, The finite element method for elliptic equations with discontinuous coefficients, Computing 5 (1970), 207–213. 10.1007/BF02248021Search in Google Scholar

[4] J. H. Bramble and J. T. King, A finite element method for interface problems in domains with smooth boundaries and interfaces, Adv. Comput. Math. 6 (1996), no. 2, 109–138. 10.1007/BF02127700Search in Google Scholar

[5] J. H. Bramble and J. E. Pasciak, New convergence estimates for multigrid algorithms, Math. Comp. 49 (1987), no. 180, 311–329. 10.1090/S0025-5718-1987-0906174-XSearch in Google Scholar

[6] J. H. Bramble and J. E. Pasciak, The analysis of smoothers for multigrid algorithms, Math. Comp. 58 (1992), no. 198, 467–488. 10.1090/S0025-5718-1992-1122058-0Search in Google Scholar

[7] J. H. Bramble, J. E. Pasciak and J. Xu, The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms, Math. Comp. 56 (1991), no. 193, 1–34. 10.1090/S0025-5718-1991-1052086-4Search in Google Scholar

[8] A.-P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. 10.4064/sm-24-2-113-190Search in Google Scholar

[9] L. Chen, H. Wei and M. Wen, An interface-fitted mesh generator and virtual element methods for elliptic interface problems, J. Comput. Phys. 334 (2017), 327–348. 10.1016/j.jcp.2017.01.004Search in Google Scholar

[10] S.-H. Chou, D. Y. Kwak and K. T. Wee, Optimal convergence analysis of an immersed interface finite element method, Adv. Comput. Math. 33 (2010), no. 2, 149–168. 10.1007/s10444-009-9122-ySearch in Google Scholar

[11] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Class. Appl. Math. 40, Society for Industrial and Applied Mathematics, Philadelphia, 2002. 10.1137/1.9780898719208Search in Google Scholar

[12] R. D. Falgout, An introduction to algebraic multigrid computing, Comput. Sci. Eng. 8 (2006), 24–33. 10.1109/MCSE.2006.105Search in Google Scholar

[13] W. Hackbusch, Multigrid Methods and Applications, Springer Ser. Comput. Math. 4, Springer, Berlin, 1985. 10.1007/978-3-662-02427-0Search in Google Scholar

[14] R. Hiptmair and J. Xu, Nodal auxiliary space preconditioning in 𝐇 ⁢ ( 𝐜 ⁢ u ⁢ r ⁢ l ) and 𝐇 ⁢ ( div ) spaces, SIAM J. Numer. Anal. 45 (2007), no. 6, 2483–2509. 10.1137/060660588Search in Google Scholar

[15] X. Hu, W. Liu, G. Qin, J. Xu and Z. Zhang, Development of a fast auxiliary subspace pre-conditioner for numerical reservoir simulators, SPE Reservoir Characterisation and Simulation Conference and Exhibition, Society of Petroleum Engineers, Texas (2011), SPE-148388-MS. 10.2118/148388-MSSearch in Google Scholar

[16] J. Huang and J. Zou, Some new a priori estimates for second-order elliptic and parabolic interface problems, J. Differential Equations 184 (2002), no. 2, 570–586. 10.1006/jdeq.2001.4154Search in Google Scholar

[17] G. Jo and D. Y. Kwak, An IMPES scheme for a two-phase flow in heterogeneous porous media using a structured grid, Comput. Methods Appl. Mech. Engrg. 317 (2017), 684–701. 10.1016/j.cma.2017.01.005Search in Google Scholar

[18] G. Jo and D. Y. Kwak, Geometric multigrid algorithms for elliptic interface problems using structured grids, Numer. Algorithms 81 (2019), no. 1, 211–235. 10.1007/s11075-018-0544-9Search in Google Scholar

[19] D. Y. Kwak and J. Lee, A modified P 1 -immersed finite element method, Int. J. Pure Appl. Math. 104 (2015), 471–494. 10.12732/ijpam.v104i3.14Search in Google Scholar

[20] D. Y. Kwak, K. T. Wee and K. S. Chang, An analysis of a broken P 1 -nonconforming finite element method for interface problems, SIAM J. Numer. Anal. 48 (2010), no. 6, 2117–2134. 10.1137/080728056Search in Google Scholar

[21] C. O. Lee, A nonconforming multigrid method using conforming subspaces, Proceedings of the Sixth Copper Mountain Conference on Multigrid Methods, NASA, Hampton (1993), 317–330. Search in Google Scholar

[22] Z. Li, T. Lin, Y. Lin and R. C. Rogers, An immersed finite element space and its approximation capability, Numer. Methods Partial Differential Equations 20 (2004), no. 3, 338–367. 10.1002/num.10092Search in Google Scholar

[23] Z. Li, T. Lin and X. Wu, New Cartesian grid methods for interface problems using the finite element formulation, Numer. Math. 96 (2003), no. 1, 61–98. 10.1007/s00211-003-0473-xSearch in Google Scholar

[24] S. F. McCormick, Multigrid Methods, Front. Appl. Math. 3, Society for Industrial and Applied Mathematics, Philadelphia, 1987. 10.1137/1.9781611971057Search in Google Scholar

[25] J. A. Roĭtberg and Z. Šeftel’, A theorem on homeomorphisms for elliptic systems and its applications, Sb. Math. 7 (1969), 439–465. 10.1070/SM1969v007n03ABEH001099Search in Google Scholar

[26] K. Stüben, A review of algebraic multigrid, J. Comput. Appl. Math. 1–2 (2001), 281–309. 10.1016/S0377-0427(00)00516-1Search in Google Scholar

[27] P. Vaněk, M. Brezina and J. Mandel, Convergence of algebraic multigrid based on smoothed aggregation, Numer. Math. 88 (2001), no. 3, 559–579. 10.1007/s211-001-8015-ySearch in Google Scholar

[28] J. Xu, The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing 56 (1996), 215–235. 10.1007/BF02238513Search in Google Scholar