An alternating-direction iteration method for Helmholtz problems

Jim Douglas; Jeffrey L. Hensley; Jean Elizabeth Roberts

Applications of Mathematics (1993)

  • Volume: 38, Issue: 4-5, page 289-300
  • ISSN: 0862-7940

Abstract

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An alternating-direction iterative procedure is described for a class of Helmholz-like problems. An algorithm for the selection of the iteration parameters is derived; the parameters are complex with some having positive real part and some negative, reflecting the noncoercivity and nonsymmetry of the finite element or finite difference matrix. Examples are presented, with an applications to wave propagation.

How to cite

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Douglas, Jim, Hensley, Jeffrey L., and Roberts, Jean Elizabeth. "An alternating-direction iteration method for Helmholtz problems." Applications of Mathematics 38.4-5 (1993): 289-300. <http://eudml.org/doc/15756>.

@article{Douglas1993,
abstract = {An alternating-direction iterative procedure is described for a class of Helmholz-like problems. An algorithm for the selection of the iteration parameters is derived; the parameters are complex with some having positive real part and some negative, reflecting the noncoercivity and nonsymmetry of the finite element or finite difference matrix. Examples are presented, with an applications to wave propagation.},
author = {Douglas, Jim, Hensley, Jeffrey L., Roberts, Jean Elizabeth},
journal = {Applications of Mathematics},
keywords = {noncoercive nonsymmetric problems; Helmholtz equation; finite difference; alternating-direction iteration method; time-stepping method; convergence; numerical examples; noncoercive nonsymmetric problems; Helmholtz equation; finite difference; alternating-direction iteration method; time-stepping method; convergence; numerical examples},
language = {eng},
number = {4-5},
pages = {289-300},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {An alternating-direction iteration method for Helmholtz problems},
url = {http://eudml.org/doc/15756},
volume = {38},
year = {1993},
}

TY - JOUR
AU - Douglas, Jim
AU - Hensley, Jeffrey L.
AU - Roberts, Jean Elizabeth
TI - An alternating-direction iteration method for Helmholtz problems
JO - Applications of Mathematics
PY - 1993
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 38
IS - 4-5
SP - 289
EP - 300
AB - An alternating-direction iterative procedure is described for a class of Helmholz-like problems. An algorithm for the selection of the iteration parameters is derived; the parameters are complex with some having positive real part and some negative, reflecting the noncoercivity and nonsymmetry of the finite element or finite difference matrix. Examples are presented, with an applications to wave propagation.
LA - eng
KW - noncoercive nonsymmetric problems; Helmholtz equation; finite difference; alternating-direction iteration method; time-stepping method; convergence; numerical examples; noncoercive nonsymmetric problems; Helmholtz equation; finite difference; alternating-direction iteration method; time-stepping method; convergence; numerical examples
UR - http://eudml.org/doc/15756
ER -

References

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  1. Douglas J., Jr., On the numerical integration of u x x + u y y = u t by implicit methods, J. Soc. Indust. Appl. Math. 3(1955), 42-65. (1955) MR0071875
  2. Douglas J., Jr., 10.1007/BF01386295, Numerische Mathematik 4 (1962), 41-63. (1962) Zbl0104.35001MR0136083DOI10.1007/BF01386295
  3. Douglas J., Jr., Dupont T., Alternating-direction Galerkin methods on rectangles, Numerical Solution of Partial Differential Equations II (Burt Hubbard, ed.), Academic Press, New York, 1971, pp. 133-214. (1971) Zbl0239.65088MR0273830
  4. Douglas J., Jr., Gunn J. E., 10.1007/BF01386093, Numerische Mathematik 6 (1964), 428-453. (1964) Zbl0141.33103MR0176622DOI10.1007/BF01386093
  5. Douglas J., Jr., Peaceman D. W., Numerical solution of two dimensional heat flow problems, A.I.Ch.E. Jour 1 (1955), 505-512. (1955) 
  6. Douglas J., Jr., Rachford H. H., Jr., 10.1090/S0002-9947-1956-0084194-4, Trans. Amer. Math. Soc. 82 (1956), 421-439. (1956) Zbl0070.35401MR0084194DOI10.1090/S0002-9947-1956-0084194-4
  7. Douglas J., Jr., Santos J. E., Sheen D., Bennethum L. S., Frequency domain treatment of one-dimensional scalar waves, Mathematical Models and Methods in Applied Sciencis (1993), to appear. (1993) Zbl0783.65070MR1212938
  8. Peaceman D. W., 10.1137/0103003, J. Soc. Ind. Appl. Math. 3 (1955), 28-41. (1955) MR0071874DOI10.1137/0103003
  9. Pearcy C. M., 10.1007/BF01386310, Numerische Mathematik 4 (1962), 172-176. (1962) Zbl0112.34802MR0145677DOI10.1007/BF01386310

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