Computational experiments and techniques for the penalty method with extrapolation

King, J. Thomas; Serbin, Steven M.

doi:10.1090/S0025-5718-1978-0471866-0

Computational experiments and techniques for the penalty method with extrapolation
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by J. Thomas King and Steven M. Serbin PDF

Math. Comp. 32 (1978), 111-126 Request permission

Abstract:

In this note we present results of a computational investigation of the extrapolated penalty method for approximate solution of elliptic boundary value problems. We investigate the effects of extrapolation and present an iterative technique for solving the extra linear algebraic systems necessary to perform the process. We indicate how convergence of the iterative procedure may be accelerated when boundary weights are appropriately selected. We consider the Euclidean relative error in the iterative procedure and the effect of conditioning. We develop a bound for the difference between an extrapolate obtained assuming exact solution of all linear systems and the corresponding quantity computed by a terminated iterative procedure.

References

Calcólo

Ivo Babuška, The finite element method with penalty, Math. Comp. 27 (1973), 221–228. MR 351118, DOI 10.1090/S0025-5718-1973-0351118-5
James R. Bunch and Donald J. Rose, Partitioning, tearing and modification of sparse linear systems, J. Math. Anal. Appl. 48 (1974), 574–593. MR 353641, DOI 10.1016/0022-247X(74)90179-6
Joel N. Franklin, Matrix theory, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1968. MR 0237517
Alan George, Nested dissection of a regular finite element mesh, SIAM J. Numer. Anal. 10 (1973), 345–363. MR 388756, DOI 10.1137/0710032
P. E. Gill, G. H. Golub, W. Murray, and M. A. Saunders, Methods for modifying matrix factorizations, Math. Comp. 28 (1974), 505–535. MR 343558, DOI 10.1090/S0025-5718-1974-0343558-6
D. Goldfarb, Modification methods for inverting matrices and solving systems of linear algebraic equations, Math. Comp. 26 (1972), 829–852. MR 317527, DOI 10.1090/S0025-5718-1972-0317527-4

The Lanczos Algorithm for the Symmetric

Problem

Eugene Isaacson and Herbert Bishop Keller, Analysis of numerical methods, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0201039
J. T. King, New error bounds for the penalty method and extrapolation, Numer. Math. 23 (1974), 153–165. MR 400742, DOI 10.1007/BF01459948
J. Th. King, A quasioptimal finite element method for elliptic interface problems, Computing 15 (1975), no. 2, 127–135 (English, with German summary). MR 405885, DOI 10.1007/BF02252861
J. Th. King and St. M. Serbin, Boundary flux estimates for elliptic problems by the perturbed variational method, Computing 16 (1976), no. 4, 339–347 (English, with German summary). MR 418485, DOI 10.1007/BF02252082
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
R. S. Martin and J. H. Wilkinson, Handbook Series Linear Algebra: Symmetric decomposition of positive definite band matrices, Numer. Math. 7 (1965), no. 5, 355–361. MR 1553944, DOI 10.1007/BF01436248
Steven M. Serbin, Computational investigations of least-squares type methods for the approximate solution of boundary value problems, Math. Comp. 29 (1975), 777–793. MR 391542, DOI 10.1090/S0025-5718-1975-0391542-X
Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502
S. D. Conte, Elementary numerical analysis: An algorithmic approach, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1965. MR 0202267