Computational experiments and techniques for the penalty method with extrapolation

Authors:
J. Thomas King and Steven M. Serbin

Journal:
Math. Comp. **32** (1978), 111-126

MSC:
Primary 65N15

DOI:
https://doi.org/10.1090/S0025-5718-1978-0471866-0

MathSciNet review:
471866

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**J. P. AUBIN, "Approximation des problèmes aux limites non homogènes et régularité de la convergence,"*Calcólo*, v. 6, 1969, pp. 117-139.**[2]**I. BABUŠKA, "The finite element method with penalty,"*Math. Comp.*, v. 27, 1973, pp. 221-228. MR**0351118 (50:3607)****[3]**J. BUNCH & D. ROSE, "Partitioning, tearing, and modification of sparse linear systems,"*J. Math. Anal. Appl.*, v.**48**, 1974, pp. 574-593. MR**0353641 (50:6124)****[4]**J. N. FRANKLIN,*Matrix Theory*, Prentice-Hall, Englewood Cliffs, N. J., 1968. MR**0237517 (38:5798)****[5]**A. GEORGE, "Nested dissection of a regular finite element mesh,"*SIAM J. Numer. Anal.*, v. 10, 1973, pp. 345-363. MR**0388756 (52:9590)****[6]**P. E. GILL, G. H. GOLUB, W. MURRAY & M. A. SAUNDERS, "Methods for modifying matrix factorizations,"*Math. Comp.*, v. 28, 1974, pp. 505-535. MR**0343558 (49:8299)****[7]**D. GOLDFARB, "Modification methods for inverting matrices and solving systems of linear equations,"*Math. Comp.*, v. 26, 1972, pp. 829-852. MR**0317527 (47:6074)****[8]**G. H. GOLUB, R. UNDERWOOD & J. H. WILKINSON,*The Lanczos Algorithm for the Symmetric**Problem*, Stanford Univ. Comput. Sci. Report SU 326P30-16, March 1972.**[9]**E. ISAACSON &. H. B. KELLER,*Analysis of Numerical Methods*, Wiley, New York, 1966. MR**0201039 (34:924)****[10]**J. T. KING, "New error bounds for the penalty method and extrapolation,"*Numer. Math.*, v. 23, 1974, pp. 153-165. MR**0400742 (53:4572)****[11]**J. T. KING, "A quasioptimal finite element method for elliptic interface problems,"*Computing*, v. 15, 1975, pp. 127-135. MR**0405885 (53:9677)****[12]**J. T. KING & S. M. SERBIN, "Boundary flux estimates for elliptic problems by the perturbed variational method,"*Computing*, v. 16, 1976, pp. 339-347. MR**0418485 (54:6524)****[13]**J. L. LIONS & E. MAGENES,*Problèmes aux Limites non Homogènes et Applications*, Vol. 1, Dunod, Paris, 1968. MR**0247243 (40:512)****[14]**R. S. MARTIN & J. H. WILKINSON, "Symmetric decomposition of positive definite band matrices,"*Numer. Math.*, v. 7, 1965, pp. 355-361. MR**1553944****[15]**S. M. SERBIN, "Computational investigations of least-squares type methods for the approximate solution of boundary value problems,"*Math. Comp.*, v. 29, 1975, pp. 777-793. MR**0391542 (52:12363)****[16]**R. VARGA,*Matrix Iterative Analysis*, Prentice-Hall, Englewood Cliffs, N. J., 1962. MR**0158502 (28:1725)****[17]**S. CONTE & C. de BOOR,*Elementary Numerical Analysis*:*An Algorithmic Approach*, McGraw-Hill, New York, 1972. MR**0202267 (34:2140)**

Retrieve articles in *Mathematics of Computation*
with MSC:
65N15

Retrieve articles in all journals with MSC: 65N15

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1978-0471866-0

Article copyright:
© Copyright 1978
American Mathematical Society