A multigrid waveform relaxation method for solving the poroelasticity equations

In this work, a multigrid waveform relaxation method is proposed for solving a collocated finite difference discretization of the linear Biot’s model. This gives rise to the first space–time multigrid solver for poroelasticity equations in the literature. The waveform relaxation iteration is based o...

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Bibliographic Details
Published in:Computational & applied mathematics Vol. 37; no. 4; pp. 4805 - 4820
Main Authors: Franco, S. R., Rodrigo, C., Gaspar, F. J., Pinto, M. A. V.
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.09.2018
Springer Nature B.V
Subjects:
Algebra
Applications of Mathematics
Applied physics
Computational mathematics
Computational Mathematics and Numerical Analysis
Finite difference method
Iterative methods
Mathematical Applications in Computer Science
Mathematical Applications in the Physical Sciences
Mathematical models
Mathematics
Mathematics and Statistics
Poroelasticity
Relaxation method (mathematics)
Multigrid waveform relaxation
65F10
Poroelasticity
Semi-algebraic mode analysis
Space–time grids
65M55
65M22
Vanka smoother
ISSN:0101-8205, 2238-3603, 1807-0302
Online Access:Get full text
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Summary:In this work, a multigrid waveform relaxation method is proposed for solving a collocated finite difference discretization of the linear Biot’s model. This gives rise to the first space–time multigrid solver for poroelasticity equations in the literature. The waveform relaxation iteration is based on a point-wise Vanka smoother that couples the pressure variable at a grid-point with the displacements around it. A semi-algebraic mode analysis is proposed to theoretically analyze the convergence of the multigrid waveform relaxation algorithm. This analysis is novel since it combines the semi-algebraic analysis, suitable for parabolic problems, with the non-standard analysis for overlapping smoothers. The practical utility of the method is illustrated through several numerical experiments in one and two dimensions.
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ISSN:0101-8205
2238-3603
1807-0302
DOI:10.1007/s40314-018-0603-9