A Finite Element Method with Strong Mass Conservation for Biot’s Linear Consolidation Model

An H(div) conforming finite element method for solving the linear Biot equations is analyzed. Formulations for the standard mixed method are combined with formulation of interior penalty discontinuous Galerkin method to obtain a consistent scheme. Optimal convergence rates are obtained.

[1]  M. Biot,et al.  THE ELASTIC COEFFICIENTS OF THE THEORY OF CONSOLIDATION , 1957 .

[2]  Béatrice Rivière,et al.  Discontinuous Galerkin methods for solving elliptic and parabolic equations - theory and implementation , 2008, Frontiers in applied mathematics.

[3]  Mary F. Wheeler,et al.  Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity , 2003, Numerische Mathematik.

[4]  M. Wheeler,et al.  A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity , 2008 .

[5]  M. N. Vrahatis,et al.  Ordinary Differential Equations In Theory and Practice , 1997, IEEE Computational Science and Engineering.

[6]  Andrea Toselli,et al.  Mixed hp-DGFEM for Incompressible Flows , 2002, SIAM J. Numer. Anal..

[7]  Susanne C. Brenner,et al.  Korn's inequalities for piecewise H1 vector fields , 2003, Math. Comput..

[8]  J. Kraus,et al.  Parameter-robust stability of classical three-field formulation of Biot's consolidation model , 2017, 1706.00724.

[9]  M. Fortin,et al.  Mixed Finite Element Methods and Applications , 2013 .

[10]  P. Hansbo,et al.  CHALMERS FINITE ELEMENT CENTER Preprint 2000-06 Discontinuous Galerkin Methods for Incompressible and Nearly Incompressible Elasticity by Nitsche ’ s Method , 2007 .

[11]  Son-Young Yi A coupling of nonconforming and mixed finite element methods for Biot's consolidation model , 2013 .

[12]  Son-Young Yi Convergence analysis of a new mixed finite element method for Biot's consolidation model , 2014 .

[13]  Douglas N. Arnold,et al.  Quadrilateral H(div) Finite Elements , 2004, SIAM J. Numer. Anal..

[14]  M. Biot General Theory of Three‐Dimensional Consolidation , 1941 .

[15]  L E Plansky On the management organization and procedural standardization of geologic research , 1985 .

[16]  S. I. Barry,et al.  Exact Solutions for Two-Dimensional Time-Dependent Flow and Deformation Within a Poroelastic Medium , 1999 .

[17]  D. Arnold An Interior Penalty Finite Element Method with Discontinuous Elements , 1982 .

[18]  Ricardo Ruiz-Baier,et al.  Locking-Free Finite Element Methods for Poroelasticity , 2016, SIAM J. Numer. Anal..

[19]  Mary F. Wheeler,et al.  A coupling of mixed and continuous Galerkin finite element methods for poroelasticity I: the continuous in time case , 2007 .

[20]  R. Showalter Poro-plastic filtration coupled to Stokes flow , 2005 .

[21]  David Wells,et al.  The deal.II Library, Version 8.4 , 2016, J. Num. Math..

[22]  Abimael F. D. Loula,et al.  On stability and convergence of finite element approximations of biot's consolidation problem , 1994 .