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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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