Computational and Applied Mathematics

, Volume 37, Issue 4, pp 4805–4820 | Cite as

A multigrid waveform relaxation method for solving the poroelasticity equations

  • S. R. Franco
  • C. Rodrigo
  • F. J. Gaspar
  • M. A. V. Pinto


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.


Poroelasticity Multigrid waveform relaxation Semi-algebraic mode analysis Vanka smoother Space–time grids 

Mathematics Subject Classification

65M55 65M22 65F10 


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Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018

Authors and Affiliations

  1. 1.Department of MathematicsState University of Centro-OesteIratiBrazil
  2. 2.Federal University of Paraná, Graduate Program in Numerical Methods in EngineeringCuritibaBrazil
  3. 3.IUMA and Applied Mathematics DepartmentUniversity of ZaragozaZaragozaSpain
  4. 4.CWI, Centrum Wiskunde & InformaticaAmsterdamThe Netherlands
  5. 5.Department of Mechanical EngineeringFederal University of ParanáCuritibaBrazil

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