Revista de Matemática: Teoría y Aplicaciones ISSN Impreso: 1409-2433 ISSN electrónico: 2215-3373

OAI: https://revistas.ucr.ac.cr/index.php/matematica/oai
Precondicionamiento del método LDG para la ecuación vectorial de Helmholtz
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Palabras clave

preconditioning
vector Helmholtz equation
LDG method
precondicionamiento
ecuación vectorial de Helmholtz
método LDG

Cómo citar

Alvarado, A., & Castillo, P. (2016). Precondicionamiento del método LDG para la ecuación vectorial de Helmholtz. Revista De Matemática: Teoría Y Aplicaciones, 23(2), 339–360. https://doi.org/10.15517/rmta.v23i2.25154

Resumen

Se presenta un estudio numérico de un precondicionador para la ecuación vectorial de Helmholtz; el cual se deriva de la técnica del Laplaciano desplazado. Se utiliza una nueva versión del método “Local Discontinuous Galerkin” (LDG) como técnica de discretización espacial. Se valida la escalabilidad del precondicionador mediante una serie de experimentos numéricos en dominios poliédricos y aproximaciones de alto orden en problemas de bajas frecuencias en el caso real.

https://doi.org/10.15517/rmta.v23i2.25154
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Citas

Airaksinen, T.; Heikkola, E.; Pennanen, A.; Toivanen, J. (2007). “An algebraic multigrid based shifted-Laplacian preconditioner for the Helmholtz equation”, J. Comput. Phys. 226(1): 1196–1210.

Alvarado, A. (2014) “Método LDG para la Ecuación Vectorial de Helmholtz”, Master’s thesis, Universidad de Puerto Rico at Mayagüez.

Alvarado, A.; Castillo, P. (2016) “Computational performance of LDG methods applied to time harmonic Maxwell equation in polyhedral domains”, J. Scientific Computing. 67(2): 453–474.

Bao, G.; Wei, G.W.; Zhao, S. (2004) “Numerical solution of the Helmholtz equation with high wavenumbers”, Internat. J. Numer. Methods Engrg. 59(3): 389–408.

Bayliss, A.; Goldstein, C.; Turkel, E. (1983) “An iterative method for the Helmholtz equation”, J. Comput. Phys. 49(3): 443–457.

Bermúdez, A.; Bullón, J.; Pena, F. (1998) “A finite element method for the thermoelectrical modelling of electrodes”, Commun. Numer. Meth. Engrg. 14(6): 581–593.

Brenner, S.; Li, F.; Sung, L. (2006) “A locally divergence-free nonconforming finite element method for the time-harmonic Maxwell’s equations”, Math. Comp. 76(258): 573–595.

Castillo, P. (2002) “Performance of discontinuous Galerkin methods for elliptic PDE’s”, SIAM J. Sci. Comput. 24(2): 524–547.

Castillo, P. (2010) “Stencil reduction algorithms for the Local Discontinuous Galerkin method”, Internat. J. Numer. Methods Engrg. 81: 1475–1491.

Castillo, P.; Cockburn, B.; Perugia, I.; Schötzau, D. (2000) “An a priori error analysis of the Local Discontinuous Galerkin method for elliptic problems”, SIAM J. Num. Anal. 38(5): 1676–1706.

Castillo, P.; Sequeira, F. (2013) “Computational aspects of the Local Discontinuous Galerkin method on unstructured grids in three dimensions”, Mathematical and Computer Modelling 57(9–10): 2279–2288.

Castillo, P.; Velázquez, E. (2008) “A numerical study of a semi-algebraic multilevel preconditioner for the Local Discontinuous Galerkin method”, Internat. J. Numer. Methods Engrg. 79: 255–268.

Clain, S.; Rappaz, J.; Swierkosz, M.; Touzani, R. (1993) “Numerical modeling of induction heating for two-dimensional geometries”, Mathematical Models and Methods in Applied Sciences 03(06): 805–822.

Cockburn, B.; Shu, C.W. (1998) “The Local Discontinuous Galerkin method for time-dependent convection-diffusion systems”, SIAM J. Num. Anal. 35: 2440–2463.

Demkowicz, L.; Vardapetyan, L. (1998) “Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements”, Comput. Methods Appl. Mech. Engrg. 152(1-2): 103–124.

Van der Vorst, H.A. (1992) “Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems”, SIAM J. Sci. Stat. Comput. 13(2): 631–644.

Dolean, V.; Fol, H.; Lanteri, S.; Perrussel, R. (2008) “Solution of the time-harmonic Maxwell equations using Discontinuous Galerkin methods”, J. Comput. Applied. Math. 218(2): 435–445.

El-Azab, A.; Garnich, M.; Kapoor, A. (2003) “Modeling of the electromagnetic forming of sheet metals: state-of-art an future needs”, Journal of Materials Processing Technology 142(3): 744–754.

Erlangga, Y.A.; Oosterlee, C.W.; Vuik, C. (2006) “A novel multigrid based preconditioner for heterogeneous Helmholtz problems”, SIAM J. Sci. Comput. 27(4): 1471–1492.

Erlangga, Y.A.; Vuik, C.; Oosterlee, C.W. (2004) “On the class of preconditioners for solving Helmholtz equation”, Appl. Numer. Math. 50(3-4): 409–425.

Greif, C.; Schötzau, D. (2007) “Preconditioners for the discretized time-harmonic Maxwell equation in mixed form”, Numerical Linear Algebra with Applications 14(4): 281–297.

Houston, P.; Perugia, I.; Schneebeli, A.; Schötzau, D. (2005) “Mixed Discontinuous Galerkin approximation of the Maxwell Operator: The indefinite case”, Modél. Math. Anal. Numér. 39(4): 727–753.

Houston, P.; Perugia, I.; Schneebeli, A.; Schötzau, D. (2005) “Interior penalty method for the indefinite time-harmonic Maxwell’s equations”, Numer. Math. 100(3): 485–518.

Li, D.; Greif, C.; Schötzau, D. (2012) “Parallel numerical solution of the time-harmonic Maxwell equations in mixed form”, Numerical Linear Algebra with Applications 19(3): 525–539.

Maxwell, J.C. (1865) “A dynamical theory of the electromagnetic field”, Phil. Trans. Royal Society Lond. 155: 459–512.

Maxwell, J.C. (1873) A Treatise on Electricity and Magnetism. Clarendon Press, Oxford.

Perugia, I.; Schötzau, D. (2003) “The hp-local discontinuous Galerkin method for low-frecuency time-harmonic Maxwell’s equations”, Math. Comp. 72(243): 1179–1214.

Perugia, I.; Schötzau, D.; Monk, P. (2002) “Stabilized interior penalty methods for the time-harmonic Maxwell’s equations”, Comput. Methods Appl. Mech. Engrg. 191(41-42): 4675–4697.

Reitzinger, S.; Schöberl, J. (2002) “An algebraic multigrid method for finite element discretizations with edge elements”, Numer. Linear Algebra with Applications 9(3): 223–238.

Saad, Y. (2001) Iterative Methods for Large Sparse Linear Systems, Second edition. Society for Industrial and Applied Mathematics, Philadelphia PA.

Theran, C. (2014) “Factorización incompleta IC(l, τ,m) por bloques para matrices generadas por métodos Local Discontinuous Galerkin”. Master’s thesis, Universidad de Puerto Rico at Mayagüez.

Tuncer, E.; Lee, B.T.; Islam, M.S.; Neikirk, D.P. (1994) “Quasi-static conductor loss calculations in transmission lines using a new conformal mapping technique”, IEEE Transactions on Microwave Theory and Techniques. 42(9): 1807–1815.

Zienkiewicz, O. (2000) “Achievements and some unsolved problems of the finite element method”, Internat. J. Numer. Methods Engrg. 47(1-3): 9–28.

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Derechos de autor 2016 Arlin Alvarado, Paul Castillo

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