Maximum-norm estimates for resolvents of elliptic finite element operators
Open Access
- 3 December 2002
- journal article
- Published by American Mathematical Society (AMS) in Mathematics of Computation
- Vol. 72 (244), 1597-1611
- https://doi.org/10.1090/s0025-5718-02-01488-6
Abstract
Let be a convex domain with smooth boundary in . It has been shown recently that the semigroup generated by the discrete Laplacian for quasi-uniform families of piecewise linear finite element spaces on is analytic with respect to the maximum-norm, uniformly in the mesh-width. This implies a resolvent estimate of standard form in the maximum-norm outside some sector in the right halfplane, and conversely. Here we show directly that such a resolvent estimate holds outside any sector around the positive real axis, with arbitrarily small angle. This is useful in the study of fully discrete approximations based on -stable rational functions, with small.Keywords
This publication has 20 references indexed in Scilit:
- Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global estimatesMathematics of Computation, 1998
- Resolvent Estimates for Elliptic Finite Element Operators in One DimensionMathematics of Computation, 1994
- Finite Element Interpolation of Nonsmooth Functions Satisfying Boundary ConditionsMathematics of Computation, 1990
- Some Optimal Error Estimates for Piecewise Linear Finite Element ApproximationsMathematics of Computation, 1982
- On the Quasi-Optimality in $L_\infty$ of the $\overset{\circ}{H}^1$-Projection into Finite Element Spaces*Mathematics of Computation, 1982
- A quasioptimal estimate in piecewise polynomial Galerkin approximation of parabolic problemsPublished by Springer Science and Business Media LLC ,1982
- Maximum norm stability and error estimates in parabolic finite element equationsCommunications on Pure and Applied Mathematics, 1980
- Generation of Analytic Semigroups by Strongly Elliptic OperatorsTransactions of the American Mathematical Society, 1974
- Interior Estimates for Ritz-Galerkin MethodsMathematics of Computation, 1974
- Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. ICommunications on Pure and Applied Mathematics, 1959