Abstract
In this paper, we observe an interesting phenomenon for a hybridizable discontinuous Galerkin (HDG) method for eigenvalue problems. Specifically, using the same finite element method, we may achieve both upper and lower eigenvalue bounds simultaneously, simply by the fine tuning of the stabilization parameter. Based on this observation, a high accuracy algorithm for computing eigenvalues is designed to yield higher convergence rate at a lower computational cost. Meanwhile, we demonstrate that certain type of HDG methods can only provide upper bounds. As a by-product, the asymptotic upper bound property of the Brezzi–Douglas–Marini mixed finite element is also established. Numerical results supporting our theory are given.
Funding statement: The first author is supported by National Natural Science Foundation of China (Grant No. 12401481). The second author is supported by National Natural Science Foundation of China (Grant No. 12331015).
Acknowledgements
The authors are very grateful to the anonymous referees for their valuable suggestions and comments.
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© 2025 Institute of Mathematics of the National Academy of Science of Belarus, published by De Gruyter, Berlin/Boston
Articles in the same Issue
- Frontmatter
- 8th Chinese–German Workshop on Computational and Applied Mathematics
- Contact Problems in Porous Media
- Mapped Coercivity for the Stationary Navier–Stokes Equations and Their Finite Element Approximations
- Super-Localized Orthogonal Decomposition Method for Heterogeneous Linear Elasticity
- A Note on the Quasi-Best Approximation Constant
- Guaranteed Upper Bounds for Iteration Errors and Modified Kačanov Schemes via Discrete Duality
- A Mixed Finite Element Method for Coupled Plates
- Asymptotic Preserving Semi-Implicit Scheme for the Shallow Water Equations with Non-Flat Bottom Topography and Manning Friction Term
- Computing Both Upper and Lower Eigenvalue Bounds by HDG Methods
- High Order Energy Stable Local Discontinuous Galerkin Methods for Camassa–Holm–Novikov Equations
- Tensor-Product Vertex Patch Smoothers for Biharmonic Problems
- High-Order Accurate Structure-Preserving Finite Volume Scheme for Ten-Moment Gaussian Closure Equations with Source Terms: Positivity and Well-Balancedness
- A Staggered Discontinuous Galerkin Method for the Simulation of Wave Propagation in Poroelastic Media
Articles in the same Issue
- Frontmatter
- 8th Chinese–German Workshop on Computational and Applied Mathematics
- Contact Problems in Porous Media
- Mapped Coercivity for the Stationary Navier–Stokes Equations and Their Finite Element Approximations
- Super-Localized Orthogonal Decomposition Method for Heterogeneous Linear Elasticity
- A Note on the Quasi-Best Approximation Constant
- Guaranteed Upper Bounds for Iteration Errors and Modified Kačanov Schemes via Discrete Duality
- A Mixed Finite Element Method for Coupled Plates
- Asymptotic Preserving Semi-Implicit Scheme for the Shallow Water Equations with Non-Flat Bottom Topography and Manning Friction Term
- Computing Both Upper and Lower Eigenvalue Bounds by HDG Methods
- High Order Energy Stable Local Discontinuous Galerkin Methods for Camassa–Holm–Novikov Equations
- Tensor-Product Vertex Patch Smoothers for Biharmonic Problems
- High-Order Accurate Structure-Preserving Finite Volume Scheme for Ten-Moment Gaussian Closure Equations with Source Terms: Positivity and Well-Balancedness
- A Staggered Discontinuous Galerkin Method for the Simulation of Wave Propagation in Poroelastic Media