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Zwei numerische Verfahren zur Lösung der biharmonischen Gleichung unter besonderer Berücksichtigung der Mehrgitteridee. (German) Zbl 0573.65085

Abteilung für Mathematik der Ruhr-Universität Bochum. 87 S. (1985).
We consider a mixed finite element discretization of the biharmonic problem. The numerical solution of the associated linear equations is investigated. In the first part of the paper a multilevel algorithm is described and convergence is established under the assumption of \(H^ 3\)-regularity. In the second part the original indefinite problem is transformed into a positive definite one for the unknown boundary values \(\lambda =\Delta u|_{\partial \Omega}\). This system is solved by a conjugate gradient method. Each iteration step requires the solution of two discrete Poisson equations. This is done by approximation using a multigrid algorithm. Furthermore we prove that using a suitable preconditioning method, the number of iteration steps required for a given accuracy is independent of the meshsize.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65E05 General theory of numerical methods in complex analysis (potential theory, etc.)
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