Efficient assembly of H(div) and H(curl) conforming finite elements

SIAM Journal on Scientific Computing

Volumn 31, Issue 6, 2009, Pages 4130-4151

  Rognes, Marie E ^a   Kirby, Robert C ^b   Logg, Anders ^a  

^a UNIVERSITY OF OSLO   (Norway)

^b TEXAS TECH UNIVERSITY   (United States)

Author keywords

Mixed finite element; Piola; Variational form compiler

Indexed keywords

COMPUTATION THEORY;

BASE FUNCTION; CURL-CONFORMING; GEOMETRICAL ENTITIES; KEYPOINTS; MIXED FINITE ELEMENTS; PIOLA; REPRESENTATION THEOREM; SIMPLE++; VARIATIONAL FORM; VARIATIONAL FORM COMPILER;

TENSORS;

EID: 73249133532     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/08073901X     Document Type: Article

References (36)

1. FEAP: A Finite Element Analysis Program, http://www.ce.berkeley.edu/ projects/feap/.
2. M. AINSWORTH AND J. COYLE, Hierarchic finite element bases on unstructured tetrahedral meshes, Internat. J. Numer. Methods Engrg., 58 (2003), pp. 2103-2130.
3. M. ALNÆs, A. LOGG, K.-A. MARDAL, O. SKAVHAUG, AND H. P. LANGTANGEN, UFC Specification and User Manual 1.1, http://www.fenics.org/ufc/(2008).
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. D. N. ARNOLD, R. S. FALK, AND R. WINTHER, Finite element exterior calculus, homological techniques, and applications, Acta Numer., 15 (2006), pp. 1-155.
6. D. N. ARNOLD, R. S. FALK, AND R. WINTHER, Mixed finite element methods for elasticity with weakly imposed symmetry, Math. Comp., 76 (2007), pp. 1699-1723.
7. D. N. ARNOLD AND R. WINTHER, Mixed finite elements for elasticity, Numer. Math., 92 (2002), pp. 401-419.
8. W. BANGERTH, R. HARTMANN, AND G. KANSCHAT, deal.II Differential Equations Analysis Library, http://www.dealii.org/(2006).
9. F. BREZZI, J. DOUGLAS JR., M. FORTIN, AND L. D. MARINI, Efficient rectangular mixed finiteelements in two and three space variables, RAIROModél. Math. Anal. Numér., 21 (1987), pp. 581-604.
10. F. BREZZI, J. DOUGLAS JR., AND L. D. MARINI, Two families of mixed finite elements for second order elliptic problems, Numer. Math., 47 (1985), pp. 217-235.
11. F. BREZZI AND M. FORTIN, Mixed and Hybrid Finite Element Methods, Springer Ser. Comput. Math. 15, Springer-Verlag, New York, 1991.
12. P. CASTILLO, R. RIEBEN, AND D. WHITE, FEMSTER: An object-oriented class library of highorder discrete differential forms, ACM Trans. Math. Software, 31 (2005), pp. 425-457.
13. P. G. CIARLET, Numerical Analysis of the Finite Element Method, Sémin. Math. Supér. 59, Les Presses de l'Université de Montréal, Montreal, 1976.
14. V. DOMÍNGUEZ AND F.-J. SAYAS, Algorithm 884: A simple Matlab implementation of the Argyris element, ACM Trans. Math. Software, 35 (2009), 11 pp.
15. J. HOFFMAN, J. JANSSON, C. JOHNSON, M. G. KNEPLEY, R. C. KIRBY, A. LOGG, L. R. SCOTT, AND G. N. WELLS, FEniCS, http://www.fenics.org/(2006).
16. R. C. KIRBY, Algorithm 839: FIAT, a new paradigm for computing finite element basis functions, ACM Trans. Math. Software, 30 (2004), pp. 502-516.
17. R. C. KIRBY, FIAT, http://www.fenics.org/fiat/(2006).
18. R. C. KIRBY, Optimizing FIAT with Level 3 BLAS, ACM Trans. Math. Software, 32 (2006), pp. 223-235.
19. R. C. KIRBY, M. KNEPLEY, A. LOGG, AND L. R. SCOTT, Optimizing the evaluation of finite element matrices, SIAM J. Sci. Comput., 27 (2005), pp. 741-758.
20. R. C. KIRBY AND A. LOGG, A compiler for variational forms, ACM Trans. Math. Software, 32 (2006), pp. 417-444.
21. R. C. KIRBY AND A. LOGG, Efficient compilation of a class of variational forms, ACMTrans. Math. Software, 33 (2007), 20 pp.
22. R. C. KIRBY AND A. LOGG, Benchmarking domain-specific compiler optimizations for variational forms, ACM Trans. Math. Software, 35 (2008), 18 pp.
23. R. C. KIRBY, A. LOGG, L. R. SCOTT, AND A. R. TERREL, Topological optimization of the evaluation of finite element matrices, SIAM J. Sci. Comput., 28 (2006), pp. 224-240.
24. R. C. KIRBY AND L. R. SCOTT, Geometric optimization of the evaluation of finite element matrices, SIAM J. Sci. Comput., 29 (2007), pp. 827-841.
25. A. Logg, FFC, http://www.fenics.org/ffc/(2006).
26. A. LOGG, Automating the finite element method, Arch. Comput. Methods Eng., 14 (2007), pp. 93-138.
27. K. A. MARDAL, X.-C. TAI, AND R. WINTHER, A robust finite element method for Darcy-Stokes flow, SIAM J. Numer. Anal., 40 (2002), pp. 1605-1631.
28. P. MONK, Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003.
29. 3, Numer. Math., 35 (1980), pp. 315-341.
30. 3, Numer. Math., 50 (1986), pp. 57-81.
31. K. B. ØLGAARD, A. LOGG, AND G. N. WELLS, Automated code generation for discontinuous Galerkin methods, SIAM J. Sci. Comput., 31 (2008), pp. 849-864.
32. K. B. ØLGAARD AND G. N. WELLS, Optimisations for quadrature representations of finite element tensors through automated code generation, ACM Trans. Math. Software, to appear.
33. O. PIRONNEAU, F. HECHT, A. L. HYARIC, AND K. OHTSUKA, FreeFEM, http://www.freefem.org/(2006).
34. P.-A. RAVIART AND J. M. THOMAS, Primal hybrid finite element methods for 2nd order elliptic equations, Math. Comp., 31 (1977), pp. 391-413.
35. A. Schneebeli, An H(curl; ω) Conforming FEM: Nédé lec's Element of the First Type, Technical report, 2003; also available online from http://www.dealii.org/developer/reports/nedelec/nedelec.pdf.
36. J. SCHÖBERL, NGSolve, http://www.hpfem.jku.at/ngsolve/index.html/ (2008).