On finite element methods for heterogeneous elliptic problems

International Journal of Solids and Structures

Volumn 45, Issue 25-26, 2008, Pages 6436-6450

  Loula, A F D ^a   Correa, M R ^a   Guerreiro, J N C ^a   Toledo, E M ^a  

^a LABORATÓRIO NACIONAL DE COMPUTAÇÃO CIENTÍFICA   (Brazil)

Author keywords

Darcy flow; Discontinuous Galerkin; Galerkin least squares; Heterogeneous media; Interface conditions; Stabilized mixed methods

Indexed keywords

COMPUTATIONAL MECHANICS; CURVE FITTING; FINITE ELEMENT METHOD; GALERKIN METHODS; POROUS MATERIALS; VARIATIONAL TECHNIQUES;

DARCY FLOW; DISCONTINUOUS GALERKIN; GALERKIN LEAST SQUARES; HETEROGENEOUS MEDIA; INTERFACE CONDITIONS; STABILIZED MIXED METHODS;

PROBLEM SOLVING;

EID: 54149085901     PISSN: 00207683     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ijsolstr.2008.08.005     Document Type: Article

References (36)

1. Alvarez G.B., Loula A.F.D., Dutra do Carmo E.D., and Rochinha F.A. A discontinuous finite element formulation for the Helmholtz equation. Computer Methods in Applied Mechanics and Engineering 195 (2006) 4018-4035
2. Arnold D.N. An interior penalty finite element method with discontinuous elements. SIAM Journal on Numerical Analysis 19 (1982) 742-760
3. Arnold D.N., Brezzi F., Cockburn B., and Marini D. A unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis 39 (2002) 1749-1779
4. Babǔska I. Error bounds for finite element method. Numerische Mathematik 16 (1971) 133-322
5. Barrenechea G., Franca L.P., and Valentin F. A Petrov-Galerkin enriched method: A mass conservative finite element method for the darcy equation. Computer Methods in Applied Mechanics and Engineering 196 (2007) 2449-2464
6. Bevilacqua L., Feijóo R.A., and Rojas L.F. A variational principle for the Laplace operator with application in the torsion of composite rods. International Journal of Solids Structures 10 (1974) 1091-1102
7. Brezzi F. On the existence, uniqueness and approximation of saddle point problems arising from lagrange multipliers. RAIRO Analyse numérique/Numerical Analysis 8 R-2 (1974) 129-151
8. Brezzi F., Cockburn B., Marini L.D., and Süli E. Stabilization mechanisms in discontinuous Galerkin finite element methods. Computer Methods in Applied Mechanics and Engineering 195 (2006) 3293-3310
9. Brezzi F., Douglas Jr. J., and Marini L.D. Two families of mixed finite elements for second order elliptic problems. Numerische Mathematik 47 (1985) 217-235
10. Brezzi F., and Fortin M. Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics Vol. 15 (1991), Springer-Verlag, New York
11. Brezzi F., Hughes T.J.R., Marini L.D., and Masud A. A mixed discontinuous Galerkin method for Darcy flow. SIAM Journal on Scientific Computing 22-23 (2005) 119-145
12. Ciarlet P.G. The Finite Element Method for Elliptic Problems. Studies in Mathematics and its Applications (1978), North-Holland Publishing Company
13. Cordes C., and Kinzelbach W. Continuous groundwater velocity fields and path lines in linear, bilinear and trilinear finite elements. Water Resources Research 28 11 (1992) 2903-2911
14. Correa, M.R., 2006. Stabilized Finite Element Methods for Darcy and Coupled Stokes-Darcy Flows. D.Sc. Thesis, LNCC, Petrópolis, RJ, Brazil (in Portuguese).
15. Correa M.R., and Loula A.F.D. Stabilized velocity post-processings for Darcy flow in heterogenous porous media. Communications in Numerical Methods in Engineering 23 (2007) 461-489
16. Correa M.R., and Loula A.F.D. Unconditionally stable mixed finite element methods for Darcy flow. Computer Methods in Applied Mechanics and Engineering 197 (2008) 1525-1540
17. Crumpton P.I., Shaw G.J., and Ware A.F. Discretisation and multigrid solution of elliptic equations with mixed derivative terms and strongly discontinuous coefficients. Journal of Computational Physics 116 (1995) 343-358
18. Dawson C. Coupling local discontinuous and continuous Galerkin methods for flow problems. Water Resources Research 28 (2005) 729-744
19. Douglas J., and Dupont T. Interior penalty procedures for elliptic and parabolic Galerkin methods. Lecture Notes in Physics 58 (1976) 207-216
20. Duarte A.V.C., Rochinha F.A., and Dutra do Carmo E.G. Discontinuous finite element formulations applied to cracked elastic domains. Computer Methods in Applied Mechanics and Engineering 185 (2000) 21-36
21. Durlofsky L.J. Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities. Water Resources Research 30 4 (1994) 965-973
22. Dutra do Carmo E.G., and Duarte A.V.C. New formulations and numerical analysis of discontinuous Galerkin methods. Computational and Applied Mathematics 21 (2002) 661-715
23. Franca L.P., Hughes T.J.R., Loula A.F.D., and Miranda I. A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin method. Numerische Mathematik 53 (1988) 123-141
24. Gelfand I.M., and Fomin S.V. Calculus of Variations (1963), Prentice-Hall
25. Georgoulis E.H., and Süli E. Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method. IMA Journal of Numerical Analysis 25 (2005) 205-220
26. Hughes T.J.R., Masud A., and Wan J. A stabilized mixed discontinuous Galerkin method for Darcy flow. Computer Methods in Applied Mechanics and Engineering 195 (2006) 3347-3381
27. Loula A.F.D., Hughes T.J.R., and Franca L.P. Mixed Petrov-Galerkin methods for the Timoshenko beam problem. Computer Methods in Applied Mechanics and Engineering 63 (1987) 133-154
28. Loula A.F.D., Rochinha F.A., and Murad M.A. Higher-order gradient post-processings for second-order elliptic problems. Computer Methods in Applied Mechanics and Engineering 128 (1995) 361-381
29. Loula, A.F.D., Toledo, E.M., 1988. Dual and Primal Mixed Petrov-Galerkin Finite Element Methods in Heat Transfer Problems. LNCC - Technical Report 048/88.
30. Masud A., and Hughes T.J.R. A stabilized finite element method for Darcy flow. Computer Methods in Applied Mechanics and Engineering 191 (2002) 4341-4370
31. Mosé R., Siegel P., and Ackerer P. Application of the mixed hybrid finite element approximation in a groundwater flow model: luxury or necessity?. Water Resources Research 30 11 (1994) 3001-3012
32. Oden J.T., Babǔska I., and Baumann C.E. A discontinuous hp finite element for diffusion problems. Journal of Computational Physics 146 (1998) 491-519
33. Raviart P.A., and Thomas J.M. A mixed finite element method for second order elliptic problems. Mathematical Aspects of the FEM. Lecture Notes in Mathematics vol. 606 (1977), Springer-Verlag 292-315
34. Silva R.C.C., and Loula A.F.D. Local residual error estimator and adaptive finite element analysis of Poisson's problems. Computers & Structures 80 (2002) 2027-2034
35. Toledo, E.M., 1990. New Mixed Finite Element Formulations With Post-Processing Schemes. D.Sc. Thesis, COPPE/UFRJ, Rio de Janeiro, RJ, Brazil (in Portuguese).
36. Wheeler M.F. An elliptic collocation-finite element method with interior penalties. SIAM Journal on Numerical Analysis 15 (1978) 152-161