Abstract
The Schwarz waveform relaxation (SWR) algorithms have many favorable properties and are extensively studied and investigated for solving time dependent problems mainly at a continuous level. In this paper, we consider a semi-discrete level analysis and we investigate the convergence behavior of what so-called semi-discrete SWR algorithms combined with discrete transmission conditions instead of the continuous ones. We shall target here the hyperbolic problems but not the parabolic problems that are usually considered by most of the researchers in general when investigating the properties of the SWR methods. We first present the classical overlapping semi-discrete SWR algorithms with different partitioning choices and show that they converge very slow. We then introduce optimal, optimized, and quasi optimized overlapping semi-discrete SWR algorithms using new transmission conditions also with different partitioning choices. We show that the new algorithms lead to a much better convergence through using discrete transmission conditions associated with the optimized SWR algorithms at the semi-discrete level. In the performed semi-discrete level analysis, we also demonstrate the fact that as the ratio between the overlap size and the spatial discretization size gets bigger, the convergence factor gets smaller which results in a better convergence. Numerical results and experiments are presented in order to confirm the theoretical aspects of the proposed algorithms and providing an evidence of their usefulness and their accuracy.
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© 2020 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Quasi-overlapping Semi-discrete Schwarz Waveform Relaxation Algorithms: The Hyperbolic Problem
- Numerical Solution to the 3D Static Maxwell Equations in Axisymmetric Singular Domains with Arbitrary Data
- The Gradient Discretisation Method for Linear Advection Problems
- Adaptive Mesh Refinement in 2D – An Efficient Implementation in Matlab
- A Robust Finite Element Method for Elastic Vibration Problems
- A Reduced Crouzeix–Raviart Immersed Finite Element Method for Elasticity Problems with Interfaces
- Regularized Collocation in Distribution of Diffusion Times Applied to Electrochemical Impedance Spectroscopy
- Ensemble Algorithm for Parametrized Flow Problems with Energy Stable Open Boundary Conditions
- On the Parameter Choice in the Multilevel Augmentation Method
- Numerical Approximations for the Variable Coefficient Fractional Diffusion Equations with Non-smooth Data
Artikel in diesem Heft
- Frontmatter
- Quasi-overlapping Semi-discrete Schwarz Waveform Relaxation Algorithms: The Hyperbolic Problem
- Numerical Solution to the 3D Static Maxwell Equations in Axisymmetric Singular Domains with Arbitrary Data
- The Gradient Discretisation Method for Linear Advection Problems
- Adaptive Mesh Refinement in 2D – An Efficient Implementation in Matlab
- A Robust Finite Element Method for Elastic Vibration Problems
- A Reduced Crouzeix–Raviart Immersed Finite Element Method for Elasticity Problems with Interfaces
- Regularized Collocation in Distribution of Diffusion Times Applied to Electrochemical Impedance Spectroscopy
- Ensemble Algorithm for Parametrized Flow Problems with Energy Stable Open Boundary Conditions
- On the Parameter Choice in the Multilevel Augmentation Method
- Numerical Approximations for the Variable Coefficient Fractional Diffusion Equations with Non-smooth Data