Dimensionally Consistent Preconditioning for Saddle-Point Problems
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Roland Herzog
Abstract
The preconditioned iterative solution of large-scale saddle-point systems is of great importance in numerous application areas, many of them involving partial differential equations. Robustness with respect to certain problem parameters is often a concern, and it can be addressed by identifying proper scalings of preconditioner building blocks. In this paper, we consider a new perspective to finding effective and robust preconditioners. Our approach is based on the consideration of the natural physical units underlying the respective saddle-point problem. This point of view, which we refer to as dimensional consistency, suggests a natural combination of the parameters intrinsic to the problem. It turns out that the scaling obtained in this way leads to robustness with respect to problem parameters in many relevant cases. As a consequence, we advertise dimensional consistency based preconditioning as a new and systematic way to designing parameter robust preconditoners for saddle-point systems arising from models for physical phenomena.
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© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Sino–German Computational and Applied Mathematics
- An Adaptive Finite Element Scheme for the Hellinger–Reissner Elasticity Mixed Eigenvalue Problem
- Adaptive Directional Compression of High-Frequency Helmholtz Boundary Element Matrices
- Hierarchical Argyris Finite Element Method for Adaptive and Multigrid Algorithms
- On the Threshold Condition for Dörfler Marking
- An Unfitted dG Scheme for Coupled Bulk-Surface PDEs on Complex Geometries
- Dimensionally Consistent Preconditioning for Saddle-Point Problems
- An Optimal Multilevel Method with One Smoothing Step for the Morley Element
- Kernel Embedding Based Variational Approach for Low-Dimensional Approximation of Dynamical Systems
- 𝑯(curl 2)-Conforming Spectral Element Method for Quad-Curl Problems
- Defects in Active Nematics – Algorithms for Identification and Tracking
- Dual-Weighted Residual A Posteriori Error Estimates for a Penalized Phase-Field Slit Discontinuity Problem
- Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods
- Error Analysis of an Unconditionally Energy Stable Local Discontinuous Galerkin Scheme for the Cahn–Hilliard Equation with Concentration-Dependent Mobility
Artikel in diesem Heft
- Frontmatter
- Sino–German Computational and Applied Mathematics
- An Adaptive Finite Element Scheme for the Hellinger–Reissner Elasticity Mixed Eigenvalue Problem
- Adaptive Directional Compression of High-Frequency Helmholtz Boundary Element Matrices
- Hierarchical Argyris Finite Element Method for Adaptive and Multigrid Algorithms
- On the Threshold Condition for Dörfler Marking
- An Unfitted dG Scheme for Coupled Bulk-Surface PDEs on Complex Geometries
- Dimensionally Consistent Preconditioning for Saddle-Point Problems
- An Optimal Multilevel Method with One Smoothing Step for the Morley Element
- Kernel Embedding Based Variational Approach for Low-Dimensional Approximation of Dynamical Systems
- 𝑯(curl 2)-Conforming Spectral Element Method for Quad-Curl Problems
- Defects in Active Nematics – Algorithms for Identification and Tracking
- Dual-Weighted Residual A Posteriori Error Estimates for a Penalized Phase-Field Slit Discontinuity Problem
- Fast Tensor Product Schwarz Smoothers for High-Order Discontinuous Galerkin Methods
- Error Analysis of an Unconditionally Energy Stable Local Discontinuous Galerkin Scheme for the Cahn–Hilliard Equation with Concentration-Dependent Mobility
