## Analysis of a discontinuous Galerkin method for the Biot’s consolidation problem.(English)Zbl 1290.74038

Summary: A fully discrete stabilized discontinuous Galerkin method is proposed to solve the Biot’s consolidation problem. The existence and uniqueness of the finite element solution are obtained. The stability of the fully discrete solution is discussed. The corresponding error estimates for the approximation of displacement and pressure in a mesh dependent norm are obtained. The error estimate for the approximation of pressure in $$L^2$$ norm is also obtained.

### MSC:

 74S05 Finite element methods applied to problems in solid mechanics 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
Full Text:

### References:

 [1] Arnold, D. N., An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19, 742-760, (1982) · Zbl 0482.65060 [2] Arnold, D. N.; Brezzi, F.; Cockburn, B.; Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 1749-1779, (2002) · Zbl 1008.65080 [3] Babus̆ka, I., The finite element method with Lagrangian multiplier, Numer. Math., 20, 179-192, (1973) · Zbl 0258.65108 [4] Riviére, B.; Shaw, S.; Wheeler, M. F.; Whiteman, J. R., Discontinuous Galerkin finite element methods for linear elasticity and quasistatic linear viscoelasticity, Numer. Math., 95, 347-376, (2003) · Zbl 1253.74114 [5] Baumann, C. E.; Oden, J. T., A discontinuous hp finite element method for convection-diffusion problems, Comput. Methods Appl. Mech. Eng., 175, 3-4, 311-341, (1999) · Zbl 0924.76051 [6] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO Anal. Numer., 8, 129-151, (1974) · Zbl 0338.90047 [7] Breezi, F.; Douglas, J., Stabilized mixed methods for the Stokes problem, Numer. Math., 53, 225-235, (1988) · Zbl 0669.76052 [8] Breezi, F.; Manzini, G.; Marini, D.; Pietra, P.; Russo, A., Discontinuous Galerkin approximation for elliptic problems, Numer. Methods Part. Diff. Eq., 16, 365-378, (2000) · Zbl 0957.65099 [9] Brooks, A. N.; Hughes, T. J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equation, Comput. Methods Appl. Mech. Eng., 32, (1982), 1999-259 · Zbl 0497.76041 [10] Cockburn, B.; Kanschat, G.; Schötzau, D.; Schwab, C., Local discontinuous Galerkin methods for the Stokes system, SIAM J. Numer. Anal., 40, 319-343, (2002) · Zbl 1032.65127 [11] Cockburn, B.; Lin, S. Y.; Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems, J. Comput. Phys., 84, 90-113, (1989) · Zbl 0677.65093 [12] Cockburn, B.; Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Math. Comput., 52, 411-435, (1989) · Zbl 0662.65083 [13] Cockburn, B.; Shu, C. W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws V: multidimensional systems, J. Comput. Phys., 144, 199-224, (1998) [14] Douglas, J.; Wang, J. P., An absolutely stabilized finite element for the Stokes problem, Math. Comput., 52, 495-508, (1989) · Zbl 0669.76051 [15] Girault, V.; Raviart, P. A., Finite element approximation of the Navier-Stokes equations, (1979), Springer-Verlag-Berlin Heidelberg New York · Zbl 0413.65081 [16] Girault, V.; Riviére, B.; Wheeler, M. F., A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and Navier-Stokes problems, Math. Comput., 74, 53-84, (2004) · Zbl 1057.35029 [17] Hwang, C. T.; Morgenstern, N. R.; Murray, D. W., On solution of plane strain consolidation problems by finite element methods, Can. Geotech. J., 8, 109-118, (1971) [18] Hu, C. Q.; Shu, C. W., A discontinuous Galerkin finite element method for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21, 666-690, (1999) · Zbl 0946.65090 [19] Luo, Y.; Feng, M. F., Discontinuous finite element methods for the Stokes equations, Math. Numer. Sinica, 28, 163-174, (2006) [20] Luo, Y.; Feng, M. F., A discontinuous element pressure gradient stabilizations for the Stokes equations based on local projections, Math. Numer. Sinica, 30, 25-36, (2008) · Zbl 1164.65481 [21] Luo, Y.; Feng, M. F., Discontinuous element pressure gradient stabilizations for the compressible Navier-Stokes equations based on local projections, Appl. Math. Mech., 29, 157-168, (2008) [22] Murad, M. A.; Loula, A. F.D., Improved accuracy in finite element analysis of biot’s consolidation problem, Comput. Methods Appl. Mech. Eng., 95, 359-382, (1992) · Zbl 0760.73068 [23] Murad, M. A.; Thomée, V.; Loula, A. F.D., Asymptotic behavior of semi discrete finite-element approximations of biot’s consolidation problem, SIAM J. Numer. Anal., 33, 1065-1083, (1996) · Zbl 0854.76053 [24] Nečas, J., Equations aux Dérivées partielles, (1965), Presses de l’Université de Montréal Montréal, Canada [25] Di Pietro, D. A.; Ern, A., Discrete functional analysis tools for discontinuous Galerkin methods with application to the incompressible Navier-Stokes equations, Math. Comput., 79, 1303-1330, (2010) · Zbl 1369.76024 [26] W.H. Reed, T.R. Hill, Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, 1973. [27] Ewing, R. E.; Wang, J. P.; Yang, Y. J., A stabilized discontinuous finite element method for elliptic problems, Numer. Linear Algebra Appl., 10, 83-104, (2003) · Zbl 1071.65551 [28] Ye, X., Discontinuous stable elements for the incompressible flow, Adv. Comput. Math., 20, 333-345, (2004) · Zbl 1040.76039 [29] Ye, X.; Xu, C. Y., A discontinuous Galerkin method for the Reissner-Mindlin plate in the primitive variables, Appl. Math. Comput., 149, 65-82, (2004) · Zbl 1044.74045 [30] Yokoo, Y.; Yamagata, K.; Nagaoka, H., Variational principles for consolidation, Soils Found., 11, 25-36, (1971) [31] Yokoo, Y.; Yamagata, K.; Nagaoka, H., Finite element methods applied to biot’s consolidation theory, Soils Found., 11, 29-46, (1971) [32] Zhou, T. X.; Feng, M. F., A least squares Petrov-Galerkin finite element for the stationary Navier-Stokes equations, Math. Comput., 60, 531-543, (1993) · Zbl 0778.65081
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.