|
Smoothed projections in finite element exterior calculus
Author(s):
Snorre
H.
Christiansen;
Ragnar
Winther.
Journal:
Math. Comp.
77
(2008),
813-829.
MSC (2000):
Primary 65N30, 53A45
Posted:
December 20, 2007
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
The development of smoothed projections, constructed by combining the canonical interpolation operators defined from the degrees of freedom with a smoothing operator, has proved to be an effective tool in finite element exterior calculus. The advantage of these operators is that they are  bounded projections, and still they commute with the exterior derivative. In the present paper we generalize the construction of these smoothed projections, such that also non-quasi-uniform meshes and essential boundary conditions are covered. The new tool introduced here is a space-dependent smoothing operator that commutes with the exterior derivative.
References:
-
- 1.
- D. N. Arnold, Differential complexes and numerical stability, in Proceedings of the International Congress of Mathematicians, Vol. I, Higher Ed. Press, Beijing, 2002, pp. 137-157. MR 1989182 (2004h:65115)
- 2.
- D. N. Arnold, R. S. Falk and R. Winther, Differential complexes and stability of finite element methods. I: The de Rham complex, in Compatible Spatial Discretizations, Vol. 142 of The IMA Volumes in Mathematics and its Applications, Springer, Berlin, 2006, pp. 23-46. MR 2256572 (2007c:65004)
- 3.
- D. N. Arnold, R. S. Falk and R. Winther, Differential complexes and stability of finite element methods. II: The elasticity complex, in Compatible Spatial Discretizations, Vol. 142 of The IMA Volumes in Mathematics and its Applications, Springer, Berlin, 2006, pp. 47-68. MR 2249345
- 4.
- D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numerica 15 (2006), pp. 1-155. MR 2269741
- 5.
- A. Bossavit, Whitney forms: A class of finite elements for three-dimensional computations in electromagnetism, IEE Trans. Mag. 135, Part A (1988), pp. 493-500.
- 6.
- S. H. Christiansen, Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimensions, Numer. Math. 107 (2007), pp. 87-106.
- 7.
- J. Dodziuk and V. K. Patodi, Riemannian structures and triangulations of manifolds, Journal of the Indian Math. Soc. 40 (1976), pp. 1-52. MR 0488179 (58:7742)
- 8.
- R. Hiptmair, Canonical construction of finite elements, Math. Comp. 68 (1999), pp. 1325-1346. MR 1665954 (2000b:65214)
- 9.
- R. Hiptmair, Higher order Whitney forms, in Geometrical methods in computational electromagnetics (F. Teixeira, ed.), Vol. 32 of PIER, EMW Publishing, Cambridge, MA, pp. 271-299.
- 10.
- R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer. 11 (2002), pp. 237-339. MR 2009375 (2004k:78028)
- 11.
- J.-C. Nédélec, Mixed finite elements in
 , Numer. Math. 35 (1980), pp. 315-341. MR 592160 (81k:65125) - 12.
- J.-C. Nédélec, A new family of mixed finite elements in
 , Numer. Math. 50 (1986), pp. 57-81. MR 864305 (88e:65145) - 13.
- J. Schöberl, A multilevel decomposition result in
 , Proceedings from the 8th European Multigrid, Multilevel, and Multiscale Conference, Scheveningen, The Hague, 2005. - 14.
- J. Schöberl, A posteriori error estimates for Maxwell equations, RICAM-Report No. 2005-10. To appear in Math. Comp. (2008)
- 15.
- H. Whitney, Geometric integration theory, Princeton University Press, 1957. MR 0087148 (19:309c)
Similar Articles:
Retrieve articles in Mathematics of Computation
with MSC
(2000):
65N30, 53A45
Retrieve articles in all Journals with MSC
(2000):
65N30, 53A45
Additional Information:
Snorre
H.
Christiansen
Affiliation:
Centre of Mathematics for Applications and Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway
Email:
snorrec@cma.uio.no
Ragnar
Winther
Affiliation:
Centre of Mathematics for Applications and Department of Informatics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway
Email:
ragnar.winther@cma.uio.no
DOI:
10.1090/S0025-5718-07-02081-9
PII:
S 0025-5718(07)02081-9
Keywords:
Exterior calculus,
finite elements,
interpolation operators
Received by editor(s):
December 20, 2006
Received by editor(s) in revised form:
March 22, 2007
Posted:
December 20, 2007
Additional Notes:
This research was supported by the Norwegian Research Council. The first author acknowledges that this work, conducted as part of the award ``Numerical analysis and simulations of geometric wave equations'' made under the European Heads of Research Councils and European Science Foundation EURYI (European Young Investigator) Awards scheme, was supported by funds from the Participating Organizations of EURYI and the EC Sixth Framework Program.
Copyright of article:
Copyright
2007,
American Mathematical Society
|
Comments: webmaster@ams.org