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Some multilevel methods on graded meshes. (English) Zbl 0996.65136

H. Yserentant’s hierarchical basis method and multilevel diagonal scaling methods [Numer. Math. 49, 379-412 (1986; Zbl 0608.65065)] on a class of refined meshes for the approximation of boundary value problems in the presence of singularities are considered. Bounds for the condition numbers of the stiffness matrix and the iteration operator are derived. It is deduced that the condition number of the BPX iteration operator is bounded by \(\ln(1/h)\). Finally, graded meshes fulfilling the general conditions are presented and numerical tests are given which confirm the theoretical bounds.

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Citations:

Zbl 0608.65065

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References:

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