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fourth-order; biharmonic operator; Laplace operators; Jacobi semi- iterative; Richardson; A.D.I.; fast Fourier transform; SIMD machine
The numerical solution of the model fourth-order elliptic boundary value problem in two dimensions is presented. The iterative procedure in which the biharmonic operator is splitted into two Laplace operators is used. After formulating the finite-difference approximation of the procedure, a formula for the evaluation of the transformed iteration vectors is developed. The Jacobi semi-iterative, Richardson and A.D.I. iterative Poisson solvers are applied to compute one transformed iteration vector. By the efficient use of the decomposition property of the corresponding iteration matrices, the fast Fourier transform algorithm needs to be applied twice in the evaluation of one iteration vector. The asymptotic number of operations for the sequential computation is $5n^2 log_2 n$, where $n^2$ is the number of interior grid points in the unit square. The result of$7 \ log_2 \ n$ parallel steps for the parallel computation on an SIMD machine with $n^2$ processors is so far the best one.
[1] B. L. Buzbee G. H. Golub C. W. Nielson: On direct methods for solving Poisson's equations. SIAM J. Num. Analys., Vol. 7 (1970), 627-656. DOI 10.1137/0707049 | MR 0287717
[2] B. L. Buzbee F. W. Dorr: Ths direct solution of the biharmonic equation on rectangular regions and the Poisson equation on irregular regions. SIAM J. Num. Analys., Vol. 11 (1974)753-762. MR 0362944
[3] F. W. Dorr: The direct solution of the discrete Poisson equation on a rectangle. SIAM Rev., Vol. 12 (1970), 248-263. DOI 10.1137/1012045 | MR 0266447 | Zbl 0208.42403
[4] L. W. Ehrlich: Solving the biharmonic equation as coupled finite difference equations. SIAM J. Num. Analys., Vol. 8 (1971), 278-287. DOI 10.1137/0708029 | MR 0288972 | Zbl 0215.55702
[5] L. W. Ehrlich: Solving the biharmonic equation in a square: A direct versus a semidirect method. Comm. ACM, Vol. 16 (1973), 711-714. DOI 10.1145/355611.362550 | Zbl 0269.65054
[6] R. Glowinski O. Pironneau: Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIAM Rev., Vol. 21 (1979), 167-212. DOI 10.1137/1021028 | MR 0524511
[7] D. Greenspan D. Schultz: Fast finite-difference solution of biharmonic problems. Comm. ACM, Vol. 15 (1972), 347-350. DOI 10.1145/355602.361313 | MR 0314277
[8] D. Greenspan D. Schultz: Simplification and improvement of a numerical method for Navier-Stokes problems. Proc. 15. Differential equations Keszthely (1975) 201 - 222. MR 0502088
[9] M. M. Gupta: Discretization error estimates for certain splitting procedures for solving first biharmonic boundary value problems. SIAM J. Num. Analys., Vol. 12. (1975), 364- 377. DOI 10.1137/0712029 | MR 0403256
[10] R. W. Hockney: The potential calculation and some applications. Methods in computational physics 9 (1970), 135-211.
[11] A. H. Sameh S. C. Chen D. J. Kuck: Parallel Poisson and biharmonic solvers. Computing, Vol. 17(1976), 219-230. MR 0438737
[12] M. Vajteršic: A fast algorithm for solving the first biharmonic boundary value problem. Computing, Vol. 23 (1979), 171-178. DOI 10.1007/BF02252095 | MR 0619928
[13] M. Vajteršic: A fast parallel solving the biharmonic boundary value problem on a rectangle. Proc. of 1st European Conference on Parallel and Distributed Processing, Toulouse 1979, 136-141.
[14] R. S. Varga: Matrix Iterative Analysis. Prentice-Hall, New York 1962. MR 0158502
[15] D. M. Young: Iterative Solution of Large Linear Systems. Academic Press, New York 1971. MR 0305568 | Zbl 0231.65034
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