×

zbMATH — the first resource for mathematics

Efficient preconditioning for the \(p\)-version finite element method in two dimensions. (English) Zbl 0754.65083
The authors study the variational problem: Find \(u\) such that \(u\in H\): \(a(u,v)=f(v)\), \(\forall v\in H\), corresponding to a second-order self- adjoint elliptic boundary value problem, where \(H\) is a Hilbert space, \(a\) is a bilinear functional defined on \(H\times H\), and \(f\) is a bounded linear functional defined over \(H\). The p-version of the finite element method is applied to this problem.
Preconditioners for a discrete variational form are formulated and analyzed and it is proved that the condition number of the preconditioned system grows as \(\log^ 2p\), where \(p\) is the degree of the finite- dimensional polynomial subspace in which the approximative finite element solution is searched.
Numerical results are presented, too, showing that the condition number grows very slowly with \(p\).

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F35 Numerical computation of matrix norms, conditioning, scaling
65F10 Iterative numerical methods for linear systems
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI