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Condition number and diagonal preconditioning: Comparison of the \(p\)-version and the spectral element methods. (English) Zbl 0858.65114
For adaptive schemes the hierarchical basis is used in the \(p\)-version of the finite element method. The condition number for the matrices of the commonly used basis introduced by I. Babuška and B. A. Szabo [Finite element analysis (1991; Zbl 0792.73003)] is investigated. For \(d\)-dimensional rectangular elements it is shown that the condition number is equivalent to \(p^{4(d-1)}\) and to \(p^{4d}\) for the stiffness and the mass matrix. Moreover, the usual diagonal preconditioning divides in the previous orders the exponents of \(p\) by two. The results are compared to those obtained for spectral elements.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65F35 Numerical computation of matrix norms, conditioning, scaling
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