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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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