A Note on Lower Bounds of Eigenvalues of Fourth Order Elliptic Operators on Local Quasi-Uniform Grids
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Youai Li
Abstract
This paper is a generalization and improvement of some recent results concerned with the lower bound property of eigenvalues produced by the Morley element of the biharmonic operator. Such an extension is of two fold. First, we generalize those results for the biharmonic operator to general fourth order elliptic operators; second, we extend them from quasi-uniform grids to local quasi-uniform grids. In particular, for the general case, we prove the asymptotic lower bound property of the discrete eigenvalues under a very weak saturation condition in terms of local meshsizes which can in general not be improved; for the special case with B = 0, we show a similar result without any assumption except the fineness of the mesh. These results can be regarded as a substantial improvement of a related result on regular meshes, including adaptive local refined meshes, for general fourth order elliptic operators, due to Yang, Li and Bi [http://arxiv.org/abs/1509.00566], while the analysis herein is completely different and largely simpler than that in the above paper.
Funding source: NSFC
Award Identifier / Grant number: 11571023
© 2016 by De Gruyter
Articles in the same Issue
- Frontmatter
- Vertex-Based Compatible Discrete Operator Schemes on Polyhedral Meshes for Advection-Diffusion Equations
- Reliable Averaging for the Primal Variable in the Courant FEM and Hierarchical Error Estimators on Red-Refined Meshes
- Monotone Finite Difference Schemes for Quasilinear Parabolic Problems with Mixed Boundary Conditions
- The Laguerre Collocation Method for Third Kind Integral Equations on Unbounded Domains
- Projection Methods for Ill-Posed Problems Revisited
- Explicit Constants in Poincaré-Type Inequalities for Simplicial Domains and Application to A Posteriori Estimates
- Regularization by Aggregation of Global and Local Data on the Sphere
- A Note on Lower Bounds of Eigenvalues of Fourth Order Elliptic Operators on Local Quasi-Uniform Grids
- Numerical Analysis and Computation of a Type of IMEX Method for the Time-Dependent Natural Convection Problem
- The Valuation of Carbon Bonds Linked with Carbon Price
Articles in the same Issue
- Frontmatter
- Vertex-Based Compatible Discrete Operator Schemes on Polyhedral Meshes for Advection-Diffusion Equations
- Reliable Averaging for the Primal Variable in the Courant FEM and Hierarchical Error Estimators on Red-Refined Meshes
- Monotone Finite Difference Schemes for Quasilinear Parabolic Problems with Mixed Boundary Conditions
- The Laguerre Collocation Method for Third Kind Integral Equations on Unbounded Domains
- Projection Methods for Ill-Posed Problems Revisited
- Explicit Constants in Poincaré-Type Inequalities for Simplicial Domains and Application to A Posteriori Estimates
- Regularization by Aggregation of Global and Local Data on the Sphere
- A Note on Lower Bounds of Eigenvalues of Fourth Order Elliptic Operators on Local Quasi-Uniform Grids
- Numerical Analysis and Computation of a Type of IMEX Method for the Time-Dependent Natural Convection Problem
- The Valuation of Carbon Bonds Linked with Carbon Price
