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On the numerical solution of the biharmonic equation and the role of squaring matrices for preconditioning. (English) Zbl 0616.65108
The mixed-variable finite element discretization of the biharmonic bidimensional problem is considered and the preconditioning of the resulting linear system by conjugate gradient methods is investigated. Previous results have been obtained by I. Gustafsson [ibid. 4, 55- 67 (1984; Zbl 0548.65083)]. By this technique each iteration step requires the solution of two discrete Poisson equations. The application of multigrid methods in order to solve numerically these equations is investigated. It seems that the use of multigrid methods has very little effect on the convergence of the conjugate gradient algorithm and hence the approximate solution does not aggravate the spectral condition number. The use of a modified incomplete Choleski decomposition proposed by I. Gustafsson [BIT 18, 142-156 (1978; Zbl 0386.65006)] to solve the Poisson equations is also discussed.
Reviewer: V.Arnăutu (Iaşi)

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
31A30 Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions
35J40 Boundary value problems for higher-order elliptic equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65F35 Numerical computation of matrix norms, conditioning, scaling
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