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Journal of Scientific Computing
Erschienen in: Journal of Scientific Computing 2/2016

23.04.2016

Robust Error Analysis of Coupled Mixed Methods for Biot’s Consolidation Model

verfasst von: Jeonghun J. Lee

Erschienen in: Journal of Scientific Computing | Ausgabe 2/2016

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Abstract

We study the a priori error analysis of finite element methods for Biot’s consolidation model. We consider a formulation which has the stress tensor, the fluid flux, the solid displacement, and the pore pressure as unknowns. Two mixed finite elements, one for linear elasticity and the other for mixed Poisson problems are coupled for spatial discretization, and we show that any pair of stable mixed finite elements is available. The novelty of our analysis is that the error estimates of all the unknowns are robust for material parameters. Specifically, the analysis does not need a uniformly positive storage coefficient, and the error estimates are robust for nearly incompressible materials. Numerical experiments illustrating our theoretical analysis are included.
Vorheriger Artikel Numerical Solution of Multidimensional Hyperbolic PDEs Using Defect Correction on Adaptive Grids
Nächster Artikel An Algorithmic Exploration of the Existence of High-Order Summation by Parts Operators with Diagonal Norm

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Literatur
1.
Anandarajah, A.: Computational Methods in Elasticity and Plasticity: Solids and Porous Media. SpringerLink: Bücher, Springer, New York (2010)MATHCrossRef
2.
Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modél. Math. Anal. Numér 19(1), 7–32 (1985)MathSciNetMATH
3.
Arnold, D.N., Brezzi, F., Douglas Jr., J.: PEERS: a new mixed finite element for plane elasticity. Jpn. J. Appl. Math. 1, 347–367 (1984)MathSciNetMATHCrossRef
4.
Arnold, D.N., Douglas Jr., J., Gupta, C.P.: A family of higher order mixed finite element methods for plane elasticity. Numer. Math. 45(1), 1–22 (1984)MathSciNetMATHCrossRef
5.
Arnold, D.N., Falk, R.S., Winther, R.: Mixed finite element methods for linear elasticity with weakly imposed symmetry. Math. Comp. 76(260), 1699–1723 (electronic) (2007)
6.
7.
8.
Boffi, D., Brezzi, F., Fortin, M.: Reduced symmetry elements in linear elasticity. Commun. Pure Appl. Anal. 8(1), 95–121 (2009)MathSciNetMATH
9.
Braess, D.: Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edn. Cambridge University Press, Cambridge (2007)MATHCrossRef
10.
Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, vol. 15. Springer, New York (1992)
11.
Brezzi, F., Douglas Jr., J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47(2), 217–235 (1985)MathSciNetMATHCrossRef
12.
Cai, Z., Starke, G.: First-order system least squares for the stress-displacement formulation: linear elasticity. SIAM J. Numer. Anal. 41(2), 715–730 (electronic) (2003)
13.
Chen, Y., Luo, Y., Feng, M.: Analysis of a discontinuous Galerkin method for the Biot’s consolidation problem. Appl. Math. Comput. 219(17), 9043–9056 (2013)MathSciNetMATH
14.
Cockburn, B., Gopalakrishnan, J., Guzmán, J.: A new elasticity element made for enforcing weak stress symmetry. Math. Comp. 79(271), 1331–1349 (2010)MathSciNetMATHCrossRef
15.
de Boer, R.: Trends in Continuum Mechanics of Porous Media, Theory and Applications of Transport in Porous Media, vol. 18. Springer, Dordrecht (2005)MATHCrossRef
16.
Fraejis de Veubeke, B.M.: Stress function approach. In: Proceedings of the World Congress on Finite Element Methods in Structural Mechanics, vol. 5, pp. J.1 – J.51 (1975)
17.
Lawrence, C.: Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998)
18.
Falk, R.S.: Finite elements for linear elasticity. In: Boffi, D., Gastaldi, L. (eds.) Mixed Finite Elements: Compatibility Conditions, vol. 1939. Springer, New York (2008)
19.
Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations, Springer Series in Computational Mathematics, vol. 5, Springer, Berlin, Theory and algorithms (1986)
20.
Gopalakrishnan, J., Guzmán, J.: A second elasticity element using the matrix bubble. IMA J. Numer. Anal. 32(1), 352–372 (2012)MathSciNetMATHCrossRef
21.
Gurtin, M.E.: An Introduction to Continuum Mechanics, Mathematics in Science and Engineering, vol. 158. Academic Press Inc., New York (1981)
22.
Guzmán, J.: A unified analysis of several mixed methods for elasticity with weak stress symmetry. J. Sci. Comput. 44(2), 156–169 (2010)MathSciNetMATHCrossRef
23.
Hairer, E., Lubich, C., Roche, M.: The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Lecture Notes in Mathematics, vol. 1409. Springer, Berlin (1989)MATH
24.
Korsawe, J., Starke, G.: A least-squares mixed finite element method for Biot’s consolidation problem in porous media. SIAM J. Numer. Anal. 43(1), 318–339 (electronic) (2005)
25.
Lee, J.J.: Towards a unified analysis of mixed methods for elasticity with weakly symmetric stress. Adv. Comput. Math. 42(2), 361–376 (2016)MathSciNetMATHCrossRef
26.
Lewis, R.W., Schrefler, B.A.: The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media. Numerical Methods in Engineering, Wiley, New Jersey (1998)MATH
27.
Logg, A., Mardal, K.-A., Wells, G.N. (eds.): Automated solution of differential equations by the finite element method, Lecture Notes in Computational Science and Engineering, vol. 84, Springer, Heidelberg, The FEniCS book (2012)
28.
Murad, M.A., Loula, A.F.D.: Improved accuracy in finite element analysis of Biot’s consolidation problem. Comput. Methods Appl. Mech. Eng. 95(3), 359–382 (1992)MathSciNetMATHCrossRef
29.
Murad, M.A., Loula, A.F.D.: On stability and convergence of finite element approximations of Biot’s consolidation problem. Int. J. Numer. Methods Eng. 37(4), 645–667 (1994)MathSciNetMATHCrossRef
30.
Murad, M.A., Thomée, V., Loula, A.F.D.: Asymptotic behavior of semidiscrete finite-element approximations of Biot’s consolidation problem. SIAM J. Numer. Anal. 33(3), 1065–1083 (1996)MathSciNetMATHCrossRef
31.
32.
33.
Phillips, P.J., Wheeler, M.F.: A coupling of mixed and continuous Galerkin finite element methods for poroelasticity. I. The continuous in time case. Comput. Geosci. 11(2), 131–144 (2007)MathSciNetMATHCrossRef
34.
Phillips, P.J., Wheeler, M.F.: A coupling of mixed and continuous Galerkin finite element methods for poroelasticity. II. The discrete-in-time case. Comput. Geosci. 11(2), 145–158 (2007)MathSciNetMATHCrossRef
35.
Phillips, P.J., Wheeler, M.F.: A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity. Comput. Geosci. 12(4), 417–435 (2008)MathSciNetMATHCrossRef
36.
Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems, mathematical aspects of finite element methods. In: Proceedings of Conference, Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975, Springer, Berlin, pp. 292–315. Lecture Notes in Math., vol. 606 (1977)
37.
38.
Stenberg, R.: On the construction of optimal mixed finite element methods for the linear elasticity problem. Numer. Math. 48(4), 447–462 (1986)MathSciNetMATHCrossRef
39.
40.
Yi, S.-Y.: A coupling of nonconforming and mixed finite element methods for Biot’s consolidation model. Numer. Methods Partial Differ. Equ. 29(5), 1749–1777 (2013)MATHCrossRef
41.
Yi, S.-Y.: Convergence analysis of a new mixed finite element method for Biot’s consolidation model. Numer. Methods Partial Differ. Equ. 30(4), 1189–1210 (2014)MATHCrossRef
42.
Yosida, K.: Functional analysis, 6th edn. Springer, Springer Classics in Mathematics, New York (1980)MATHCrossRef
Metadaten
Titel
Robust Error Analysis of Coupled Mixed Methods for Biot’s Consolidation Model
verfasst von
Jeonghun J. Lee
Publikationsdatum
23.04.2016
Verlag
Springer US
Erschienen in
Journal of Scientific Computing / Ausgabe 2/2016
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-016-0210-0

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