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Maximum norm resolvent estimates for elliptic finite element operators. (English) Zbl 0979.65097
Author’s abstract: We study a finite element approximation $$A_h$$, based on simplicial Lagrange elements, of a second-order elliptic operator $$A$$ under homogeneous Dirichlet boundary conditions in two and three dimensions, where $$h$$ is thought of as a meshsize. The main result of the paper is a new resolvent estimate for the operator $$A_h$$ in the $$L_\infty$$-norm. This estimate is uniform with respect to $$h$$ for the case with at least quadratic elements. In the case with linear elements, the estimate contains on the right a factor proportional to $$(\log\log{1\over h})^\nu$$, where $$\nu= 1$$ or $$\nu={5\over 4}$$ in two or three dimensions, respectively.

##### MSC:
 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations
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