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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 \(p\).

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
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