Improved accuracy in finite element analysis of Biot’s consolidation problem.(English)Zbl 0760.73068
The Biot’s consolidation problem consisting in a boundary value problem for coupled equations describing linear elasticity and incompressible Newtonian fluid (porous incompressible medium) is considered. To improve the rates of convergence of finite element schemes to solve the problem, the authors propose a sequential Galerkin (Petrov-Galerkin) post- processing technique and study its applicability and advantages. There are numerical illustrations of the proposed scheme.
Reviewer: L.P.Lebedev (Rostov-na-Donu)
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74E05 | Inhomogeneity in solid mechanics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
porous incompressible medium;linear elasticity;incompressible Newtonian fluid;convergence;Galerkin (Petrov-Galerkin) post-processing techniqueCite
Full Text:DOI
References:
| [1] | Terzaghi, K., Theoretical Soil Mechanics (1942), Wiley: Wiley New York |
| [2] | Biol, M., General theory of 3-D consolidation, J. Appl. Phys., 12, 155-169 (1941) ·JFM 67.0837.01 |
| [3] | Biot, M., Theory of elasticity and consolidation for a porous anistropic solid, J. Appl. Phys., 26, 182-185 (1955) ·Zbl 0067.23603 |
| [4] | Biot, M., General solutions of the equations of elasticity and consolidation for a porous material, J. Appl. Mech., 78, 91-96 (1956) ·Zbl 0074.19101 |
| [5] | Biot, M.; Willis, D. G., The elastic coefficient of the theory of consolidation, J. Appl. Mech., 79, 594-601 (1957) |
| [6] | Hwang, C. T.; Morgenstera, N. R.; Murray, D. W., On solutions of plane strain consolidation problems by finite element methods, Canad. Geotech. J., 8, 109-118 (1971) |
| [7] | Sandhu, R. S.; Wilson, E. L., Finite element analysis of seepage in elastic media, J. Engrg. Mech. Div. Amer. Soc. Civil Engrg., 95, 641-652 (1969) |
| [8] | Yokoo, Y.; Yamagata, K.; Nagaoka, H., Variational principles for consolidation, Soils and Foundations, 11, 4, 25-36 (1971) |
| [9] | Yokoo, Y.; Yamagata, K.; Nagaoka, H., Finite element method applied to Biot’s consolidation theory, Soils and Foundations, 11, 1, 29-46 (1971) |
| [10] | Babuška, I., The finite element method with Lagrangian multipliers, Numer. Math., 20, 179-192 (1973) ·Zbl 0258.65108 |
| [11] | Girault, V.; Raviart, P. A., Finite Element Methods for Navier Stokes Equations (1986), Springer: Springer Berlin ·Zbl 0396.65070 |
| [12] | Hughes, T. J.R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis (1987), Prentice Hall: Prentice Hall Englewood Cliffs, NJ ·Zbl 0634.73056 |
| [13] | Oden, J. T.; Carey, G. F., Finite Elements: Mathematical Aspects (1982), Prentice Hall: Prentice Hall Englewood Cliffs, NJ |
| [14] | Roberts, J. E.; Thomas, J. M., Mixed and hybrid finite element methods, (Rapports de Recherche, 737 (1987), INRIA) ·Zbl 0875.65090 |
| [15] | Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO Anal. Numér., 8, R-2, 129-151 (1974) ·Zbl 0338.90047 |
| [16] | Bercovier, M.; Pironneau, O., Error estimates for finite element solution of the Stokes problem in the primitive variable, Numer. Math., 33, 211-224 (1979) ·Zbl 0423.65058 |
| [17] | Fortin, M., Old and new finite elements for imcompressible flows, Internat J. Numer. Methods Fluids, 1, 347-364 (1983) ·Zbl 0467.76030 |
| [18] | Oden, J. T.; Jacquotte, O.-P., Stability of some mixed finite element methods for Stokesian flows, Comput. Methods Appl. Mech. Engrg., 43, 231-247 (1984) ·Zbl 0598.76033 |
| [19] | Stenberg, R., Analysis of mixed finite element methods for the Stokes problem: A unified approach, Math. Comput., 42, 9-23 (1984) ·Zbl 0535.76037 |
| [20] | Douglas, J.; Dupont, T., Galerkin methods for parabolic problems, SIAM J. Numer. Anal., 7, 575-625 (1970) |
| [21] | Wheeler, M. F., A priori \(L^2\) error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal., 10, 723-759 (1973) ·Zbl 0232.35060 |
| [22] | Thomée, V., Galerkin finite element methods for parabolic problems, (Lecture Notes in Mathematics 1054 (1984), Springer: Springer Berlin) ·Zbl 0617.65118 |
| [23] | Johnson, C.; Thomée, V., Error estimates for some mixed finite element methods for parabolic type problems, RAIRO Anal. Numér., 15, 41-78 (1981) ·Zbl 0476.65074 |
| [24] | Suri, M., Mixed finite element methods for the approximation of time-dependent problems, Numer. Methods Partial Differential equations, 101-111 (1986) ·Zbl 0624.65108 |
| [25] | Ženišek, A., The existence and uniqueness theorem in Biot’s consolidation theory, Apl. Mat., 29, 194-210 (1984) ·Zbl 0557.35005 |
| [26] | Murad, M. A., (DSc Thesis (1990), Pontifícia Universidade Católica do Rio de Janeiro) |
| [27] | Raviart, P. A.; Thomas, J. M., A mixed finite element method for second order elliptic problems, (Mathematical Aspects of the Finite Element Method. Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics, 606 (1975), Springer: Springer Berlin), 292-315 ·Zbl 0362.65089 |
| [28] | Arnold, D. A.; Brezzi, F.; Douglas, J., PEERS: A new mixed finite element for plane elasticity, Japan J. Appl. Math., 1, 347-367 (1984) ·Zbl 0633.73074 |
| [29] | Ciarlet, P. G., The finite element method for elliptic problems (1978), North-Holland: North-Holland Amsterdam ·Zbl 0445.73043 |
| [30] | Toledo, E. M.; Loula, A. F.D.; Guerreiro, J. N.C., Mixed Petrov-Galerkin formulations for bidimensional elasticity, (XI Latin American and Iberic Congress on Computational Methods in Engineering. XI Latin American and Iberic Congress on Computational Methods in Engineering, Porto, Portugal (1989)) ·Zbl 1168.74459 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
