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Advanced Numerical Methods in the Mathematical Sciences

Ludmil Zikatanov, Pennsylvania State Universityand Bulgarian Academy of Sciences (Bulgaria)
Stability and Monotonicity in the Low Order Discretizations of the Biot's Model

Authors

  • C. Rodrigo, Department of Mathematics, University of Zaragoza (Spain)
  • F.J. Gaspar, Department of Mathematics, University of Zaragoza (Spain)
  • X. Hu, Department of Mathematics, Tufts University
  • L.T. Zikatanov, Department of Mathematics, Pennsylvania State University, and Institute of Mathematics and Informatics, Bulgarian Academy of Sciences (Bulgaria)

Abstract

We consider finite element discretizations of the Biot's model in poroelasticity with lowest order (MINI and stabilized P1-P1) elements. We show convergence of discrete schemes which are implicit in time and use these types of elements in space. We deal with 1, 2, and 3 spatial dimensions in a unified fashion. We also address the issue related to the presence of non-physical oscillations in the pressure approximations for low permeabilities and/or small time steps. We show that even in 1D a Stokes-stable finite element pair does not provide a monotone discretization for low permeabilities. We then introduce a stabilization term which removes the oscillations. We present numerical results confirming the monotone behavior of the stabilized schemes.