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Maximum-norm estimates for resolvents of elliptic finite element operators. (English) Zbl 1028.65113
Let $$A_h$$ be the discrete Laplacian for quasi-uniform families of piecewise linear finite element spaces on a convex domain with smooth boundary. It is shown that outside any sector around the axis with arbitrarily small angle $|(\lambda I -A_h)^{-1} v|\leq C (1+|\lambda |)^{-1} |v|$ holds for the maximum norm. The proof for $$|\lambda |\leq c h^2$$ differs from that for $$|\lambda |\geq c h^2$$, and special care is done in order to avoid the log term that is connected with the Ritz projector. This direct proof of the estimate of the resolvent is independent of an earlier proof via the semigroup generated by $$A_h$$.

MSC:
 65N15 Error bounds for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 65J10 Numerical solutions to equations with linear operators (do not use 65Fxx)
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