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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 $$p$$.

##### 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
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