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Block triangular preconditioners for the discretized time-harmonic Maxwell equations in mixed form. (English) Zbl 1198.65219
Summary: We consider the solution of the saddle point linear systems arising from the finite element discretization of the time-harmonic Maxwell equations in mixed form. Two types of block triangular Schur complement-free preconditioners used with Krylov subspace methods are proposed, involving the choice of the parameter. Furthermore, we give the optimal parameter in practice. Theoretical analysis shows that all eigenvalues of the preconditioned matrices are strongly clustered. Finally, numerical experiments that validate the analysis are presented.

MSC:
65N22Solution of discretized equations (BVP of PDE)
65F08Preconditioners for iterative methods
78M10Finite element methods (optics)
References:
[1]Chen, Z. M.; Du, Q.; Zou, J.: Finite element methods with matching and non-matching meshes for Maxwell equations with discontinuous coefficients, SIAM J. Numer. anal. 37, 1542-1570 (2000) · Zbl 0964.78017 · doi:10.1137/S0036142998349977
[2]Greif, C.; Schötzau, D.: Preconditioners for the discretized time-harmonic Maxwell equations in mixed form, Numerical lin. Alg. appl. 14, 281-297 (2007) · Zbl 1199.78010 · doi:10.1002/nla.515
[3]Greif, C.; Schötzau, D.: Preconditioners for saddle point linear systems with highly singular (1,1) blocks, Etna 22, 114-121 (2006) · Zbl 1112.65042 · doi:http://etna.mcs.kent.edu/vol.22.2006/pp.dir/pp.html
[4]Hu, Q.; Zou, J.: Substructuring preconditioners for saddle-point problems arising from Maxwell’s equations in three dimensions, Math. comput. 73, 35-61 (2004) · Zbl 1048.65109 · doi:10.1090/S0025-5718-03-01541-2
[5]Perugia, I.; Schötzau, D.; Monk, P.: Stabilized interior penalty methods for the time-harmonic Maxwell equations, Comput. methods appl. Mech. engrg. 191, 4675-4697 (2002) · Zbl 1040.78011 · doi:10.1016/S0045-7825(02)00399-7
[6]Monk, P.: Finite element methods for Maxwell’s equations, (2003) · Zbl 1024.78009 · doi:10.1093/acprof:oso/9780198508885.001.0001
[7]Gopalakrishnan, J.; Pasciak, J. E.; Demkowicz, L. F.: Analysis of a multigrid algorithm for time harmonic Maxwell equations, SIAM J. Numer. anal. 42, 90-108 (2004) · Zbl 1079.78025 · doi:10.1137/S003614290139490X
[8]Nédélec, J. C.: Mixed finite elements in R3, Numer. math. 35, 315-341 (1980) · Zbl 0419.65069 · doi:10.1007/BF01396415
[9]Benzi, M.; Liu, J.: Block preconditioning for saddle point systems with indefinite (1,1) block, Int. J. Comput. math. 84, No. 8, 1117-1129 (2007) · Zbl 1123.65028 · doi:10.1080/00207160701356605
[10]Cao, Z. H.: Positive stable block triangular preconditioners for symmetric saddle point problems, Appl. numer. Math. 57, 899-910 (2007) · Zbl 1118.65021 · doi:10.1016/j.apnum.2006.08.001
[11]Klawonn, A.: Block-triangular preconditioners for saddle point problems with a penalty term, SIAM J. Sci. comput. 19, 172-184 (1998) · Zbl 0917.73069 · doi:10.1137/S1064827596303624
[12]Simoncini, V.: Block triangular preconditioners for symmetric saddle-point problems, Appl. numer. Math. 49, 63-80 (2004) · Zbl 1053.65033 · doi:10.1016/j.apnum.2003.11.012
[13]Murphy, M. F.; Golub, G. H.; Wathen, A. J.: A note on preconditioning for indefinite linear systems, SIAM J. Sci. comput. 21, 1969-1972 (2000) · Zbl 0959.65063 · doi:10.1137/S1064827599355153
[14]Silvester, D.; Wathen, A.: Fast iterative solution of stabilised Stokes systems. Part II: Using general block preconditioners, SIAM J. Numer. anal. 31, 1352-1367 (1994) · Zbl 0810.76044 · doi:10.1137/0731070
[15]Wathen, A.; Silvester, D.: Fast iterative solution of stabilised Stokes systems. Part I: Using simple diagonal preconditioners, SIAM J. Numer. anal. 30, 630-649 (1993) · Zbl 0776.76024 · doi:10.1137/0730031
[16]Golub, G. H.; Wu, X.; Yuan, J. Y.: SOR-like methods for augmented systems, Bit 41, No. 1, 71-85 (2001) · Zbl 0991.65036 · doi:10.1023/A:1021965717530