Abstract
A new class of explicit approximate inverse preconditioning is introduced for solving fourth-order equations, based on the ‘coupled equation approach’, by the domain decomposition method in conjunction with various finite difference approximation schemes. Explicit approximate inverse arrow-type matrix techniques, based on the concept of sparse L U-type factorization procedures, are introduced for computing a class of approximate inverses. Explicit preconditioned conjugate gradient-type schemes are presented for the efficient solution of linear systems. Applications of the method to a biharmonic problem are discussed and numerical results are given.
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References
Axelsson, O.: Notes on the numerical solution of biharmonic equation, J. Inst. Math. Appl. 11 (1973), 213–226.
Bjorstad, P. E. and Widlund, O.: Solving elliptic problems on regions partitioned into substructures, In: G. Birkhoff and A. Schoenstadt (eds), Elliptic Problem Solver II, Academic Press, New York, 1984, pp. 245–256.
Bramble, J. H., Pasciak, J. G. and Schatz, A. H.: The construction of preconditioners for elliptic problems by substructuring I, Math. Comput. 47 (1986), 103–134.
Bramble, J. H., Pasciak, J. G. and Schatz, A. H.: An iterative method for elliptic problems on regions partitioned in substructures, Math. Comput. 46 (1986), 361–369.
Buzbee, B. L. and Dorr, F. W.: The direct solution of the biharmonic equation on rectangular regions and the Poisson equation on irregular regions, SIAM J. Numer. Anal. 11 (1974), 753–763.
Chan, T. F.: Analysis of preconditioners for domain decomposition, SIAM J. Numer. Anal. 24 (1987), 382–390.
Dryja, M.: A finite element capacitance method for elliptic problems on regions partitioned into subregions, Numer. Math. 44 (1984), 153–168.
Dryja, M.: A capacitance matrix method for Dirichlet problem on polygonal region, Numer. Math. 39 (1982), 51–64.
Ehrlich, L. W.: Solving the biharmonic equation in a square: A direct versus a semi-direct method, Comm. ACM 16 (1973), 711–714.
Ehrlich, L. W.: Solving the biharmonic equation as a coupled finite difference equations, SIAM J. Numer. Anal. 8 (1971), 278–287.
Evans, D. J.: Preconditioning Methods: Theory and Applications, Gordon and Breach, New York, 1983.
Glowinski, R., Periaux, J., Shi, Z-C. and Widlund, O.: Domain decomposition methods in sciences and engineering, Eight International Conference, Wiley, New York, 1997.
Golub, G. H. and van Loan, C.: Matrix Computations, The Johns Hopkins University Press, 1996.
Gravvanis, G. A.: A note on the rate of convergence and complexity of domain decomposition approximate inverse preconditioning, In: K. J. Bathe (ed.), Computational Fluid and Solid Mechanics, Proc. First MIT Conference on Computational Fluid and Solid Mechanics, Vol. 2, Elsevier, Amsterdam, 2001, pp. 1586–1589.
Gravvanis, G. A.: Explicit preconditioning conjugate gradient schemes for solving biharmonic equations, Engrg. Comput. 17(2) (2000), 154–165.
Gravvanis G. A.: Explicit preconditioned generalized domain decomposition methods, I, J. Appl. Math. 4(1) (2000), 57–71.
Gravvanis G. A.: Explicit isomorphic iterative methods for solving arrow-type linear systems, I, J. Comp. Math. 74(2) (2000), 195–206.
Gravvanis, G. A.: Explicit approximate inverse finite element preconditioning for solving biharmonic equations, In: M. Ingber, H. Power, and C. A. Brebbia (eds), Applications of High Performance Computing in Engineering VI,WIT Press, 2000, pp. 411–420.
Gravvanis, G. A.: Banded approximate inverse matrix techniques, Engrg. Comput. 16(3) (1999), 337–346.
Gravvanis, G. A.: Parallel matrix techniques, In: K. D. Papailiou, D. Tsahalis, J. Periaux, C. Hirsch and M. Pandolfi (eds), Computational Fluid Dynamics 98, Vol. 1, Wiley, New York, 1998, pp. 472–477.
Gravvanis, G. A.: An approximate inverse matrix technique for arrowhead matrices, I, J. Comput. Math. 70 (1998), 35–45.
Gravvanis, G. A.: The rate of convergence of explicit approximate inverse preconditioning, I, J. Comput. Math. 60 (1996), 77–89. 192 GEORGE A. GRAVVANIS
Gravvanis, G. A.: Explicit preconditioned methods for solving 3D boundary value problems by approximate inverse finite element matrix techniques, I, J. Comput. Math. 56 (1995), 77–93.
Greenspan, D. and Schultz, D.: Fast finite difference solution of biharmonic boundary value problem, Comm. ACM 15 (1972), 347–350.
Grote, M. J. and Huckle, T.: Parallel preconditioning with sparse approximate inverses, SIAM J. Sci. Comput. 18 (1997), 838–853.
Keyes, D. E. and Xu, J.: Domain decomposition methods in scientific and engineering computing, In: Proc. Seventh International Conference on Domain Decomposition, Amer.Math. Soc., Providence, 1994.
Ortega, J. M.: Introduction to Parallel and Vector Solution of Linear Systems, Plenum Press, New York, 1989.
Schwarz, H. R.: Finite Element Methods, Academic Press, New York, 1989.
Van der Vorst, H. A. and Vuik, C.: GMRESR: A family of nested GMRES methods, Numer. Linear Algebra Appl. 1(4) (1994), 369–386.
Vuik, C. and Van der Vorst, H. A.: A comparison of some GMRES-like methods, Linear Algebra Appl. 160 (1992), 131–162.
Yousif, W. S. and Evans, D. J.: Explicit block iterative method for the solution of biharmonic equation, Numer. Meth. Partial Differential Equations 9 (1993), 1–12.
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Gravvanis, G.A. Domain Decomposition–Finite Difference Approximate Inverse Preconditioned Schemes for Solving Fourth-Order Equations. Journal of Mathematical Modelling and Algorithms 1, 181–192 (2002). https://doi.org/10.1023/A:1020538522302
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DOI: https://doi.org/10.1023/A:1020538522302