Rehabilitation of the lowest-order Raviart-Thomas element on quadrilateral grids

SIAM Journal on Numerical Analysis

Volumn 47, Issue 1, 2008, Pages 487-507

  Bochev, Pavel B ^a   Ridzal, Denis ^a  

^a SANDIA NATIONAL LABORATORIES   (United States)

Author keywords

Least squares methods; Mimetic methods; Mixed methods; Quadrilateral; Raviart Thomas

Indexed keywords

DARCY EQUATIONS; DIVERGENCE OPERATORS; FINITE ELEMENT SOLUTION; FINITE ELEMENT SPACE; GALERKIN; LEAST SQUARES METHODS; MIXED METHOD; ORDER OF CONVERGENCE; RAVIART-THOMAS;

GALERKIN METHODS; LINEAR SYSTEMS; SURFACE PHENOMENA;

FINITE ELEMENT METHOD;

EID: 77952710017     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/070704265     Document Type: Article

References (22)

1. T. ARBOGAST, C. DAWSON, P. T. KEENAN, M. F. WHEELER, AND I. YOTOV, Enhanced cellcentered finite differences for elliptic equations on general geometry, SIAM J. Sci. Comput., 19 (1998), pp. 404-425.
2. V. ARNOLD, Mathematical Methods of Classical Mechanics, Springer-Vorlag, New York, 1989.
3. D. N. ARNOLD, D. BOFFI, AND R. S. FALK, Approximation by quadrilateral finite elements, Math. Comp., 71 (2002), pp. 909-922.
4. D. N. ARNOLD, D. BOFFI, AND R. S. FALK, Quadrilateral H(div) finite elements, SIAM J. Numer. Anal. 42 (2005), pp. 2429-2451.
5. M. BERNDT, K. LIPNIKOV, M. SHASHKOV, M. F. WHEELER, AND I. YOTOV, Superconvergence of the velocity in mimetic finite difference methods on quadrilaterals, SIAM J. Numer. Anal., 43 (2005), pp. 1728-1749.
6. P. B. BOCHEV AND M. D. GUNZBURGER, Finite element methods of least-squares type, SIAM Rev., 40 (1998), pp. 789-837.
7. P. B. BOCHEV AND M. D. GUNZBURGER, On least-squares finite element for the Poisson equation and their connection to the Dirichlet and Kelvin principles, SIAM J. Numer. Anal., 43 (2005), pp. 340-362.
8. P. BOCHEV AND M. GUNZBURGER, A locally conservative least-squares method for Darcy flows, Comm. Numer. Methods Engrg., 24 (2008), pp. 97-110.
9. D. BOFFI, F. KIKUOHI, AND J. SCHBERL, Edge element computation of Maxwell's eigenvalues on general quadrilateral meshes, Math. Models Methods Appl. Sci., 16 (2006), pp. 265-273.
10. D. BOFFI AND L. GASTALDI, Some Remarks on Quadrilateral Mixed Finite Elements, submitted to the Fifth MIT Conference on Computational Fluid and Solid Mechanics.
11. F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991.
12. P. CIARLET, The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics 40, SIAM, Philadelphia, 2002.
13. G. Fix, M. GUNZBURGER, AND R. NICOLAIDES, On finite element methods of the least-squares type, Comput. Math. Appl., 5 (1979), pp. 87-98.
14. V. GIRAULT AND P. RAVIART, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer-Vorlag, Berlin, 1986.
15. J. HYMAN AND M. SHASHKOV, Natural discretizations for the divergence, gradient, and curl on logically rectangular grids, Comput. Math. Appl., 33 (1997), pp. 81-104.
16. J. HYMAN AND M. SHASHKOV, Adjoint operators for the natural discretizations of the divergence, gradient and curl on logically rectangular grids, Appl. Numer. Math., 25 (1997), pp. 413-442.
17. YU. KUZNETSOV AND S. REPIN, Convergence analysis and error estimates for mixed finite element method on distorted meshes, J. Numor. Math., 13 (2005), pp. 33-51.
18. L. G. MARGOLIN AND M. SHASHKOV, Second-order sign-preserving conservative interpolation (remapping) on general grids, J. Comput. Phys., 184 (2003), pp. 266-298.
19. P. A. RAVIART AND J. M. THOMAS, A mixed finite element method for second order elliptic problems, in Mathematical Aspects of Finite Element Method, I. Galligani and E. Magcnos, cds., Lecture Notes in Math. 606, Springer, Berlin, 1977, pp. 292-315.
20. M. SHASHKOV, Conservative Finite Difference Methods on General Grids, CRC Press, Boca Raton, FL, 1996.
21. J. SHEN, Mixed Finite Element Methods on Distorted Rectangular Grids, Technical report ISC-94-13-MATH, Texas A & M University, College Station, TX, 1994.
22. M. WHEELER, AND I. YOTOV, A cell-centered finite difference method on quadrilaterals, in Compatible Spatial Discretizations, D. Arnold, P. Bochev, R. Lehoucq, R. Nicolaides, M. Shashkov, eds., IMA Vol. Math. Appl. 142, Springer, New York, 2006, pp. 189-207.