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Multigrid methods for the biharmonic problem discretized by conforming $$C^ 1$$ finite elements on nonnested meshes. (English) Zbl 0842.65081
Authors’ abstract: We consider multigrid algorithms for the biharmonic problem discretized by conforming $$C^1$$ finite elements. Most finite elements for the biharmonic equation are nonnested in the sense that the coarse finite element space is not a subspace of the space of similar elements defined on a refined mesh. To define multigrid methods, certain intergrid transfer operators have to be constructed. We construct intergrid transfer operators that satisfy a certain stable approximation property. The so-called regularity-approximation assumption is established by using this stable approximation property of the intergrid transfer operator. Optimal convergence properties of the W-cycle and a uniform condition number estimate for the variable V-cycle preconditioner are established by applying an abstract result of the first author, J. E. Pasciak and J. Xu [Math. Comput. 55, No. 191, 1-22 (1990; Zbl 0703.65076)]. Our theory covers the cases when the multilevel triangulations are nonnested and the spaces on different levels are defined by different finite elements.

##### MSC:
 65N55 Multigrid methods; domain decomposition 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 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65F35 Numerical computation of matrix norms, conditioning, scaling
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