Smoothed projections in finite element exterior calculus

Christiansen, Snorre; Winther, Ragnar

doi:10.1090/S0025-5718-07-02081-9

Smoothed projections in finite element exterior calculus
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by Snorre H. Christiansen and Ragnar Winther PDF

Math. Comp. 77 (2008), 813-829 Request permission

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 $L^2$ 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

Douglas N. Arnold, Differential complexes and numerical stability, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 137–157. MR 1989182
Douglas N. Arnold, Pavel B. Bochev, Richard B. Lehoucq, Roy A. Nicolaides, and Mikhail Shashkov (eds.), Compatible spatial discretizations, The IMA Volumes in Mathematics and its Applications, vol. 142, Springer, New York, 2006. Papers from the IMA Hot Topics Workshop on Compatible Spatial Discretizations for Partial Differential Equations held at the University of Minnesota, Minneapolis, MN, May 11–15, 2004. MR 2256572, DOI 10.1007/0-387-38034-5
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Differential complexes and stability of finite element methods. II. The elasticity complex, Compatible spatial discretizations, IMA Vol. Math. Appl., vol. 142, Springer, New York, 2006, pp. 47–67. MR 2249345, DOI 10.1007/0-387-38034-5_{3}
Douglas N. Arnold, Richard S. Falk, and Ragnar Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer. 15 (2006), 1–155. MR 2269741, DOI 10.1017/S0962492906210018
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.
S. H. Christiansen, Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimensions, Numer. Math. 107 (2007), pp. 87–106.
J. Dodziuk and V. K. Patodi, Riemannian structures and triangulations of manifolds, J. Indian Math. Soc. (N.S.) 40 (1976), no. 1-4, 1–52 (1977). MR 488179
R. Hiptmair, Canonical construction of finite elements, Math. Comp. 68 (1999), no. 228, 1325–1346. MR 1665954, DOI 10.1090/S0025-5718-99-01166-7
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.
R. Hiptmair, Finite elements in computational electromagnetism, Acta Numer. 11 (2002), 237–339. MR 2009375, DOI 10.1017/S0962492902000041
J.-C. Nédélec, Mixed finite elements in $\textbf {R}^{3}$, Numer. Math. 35 (1980), no. 3, 315–341. MR 592160, DOI 10.1007/BF01396415
J.-C. Nédélec, A new family of mixed finite elements in $\textbf {R}^3$, Numer. Math. 50 (1986), no. 1, 57–81. MR 864305, DOI 10.1007/BF01389668
J. Schöberl, A multilevel decomposition result in $H(\mathrm {curl})$, Proceedings from the 8th European Multigrid, Multilevel, and Multiscale Conference, Scheveningen, The Hague, 2005.
J. Schöberl, A posteriori error estimates for Maxwell equations, RICAM–Report No. 2005-10. To appear in Math. Comp. (2008)
Hassler Whitney, Geometric integration theory, Princeton University Press, Princeton, N. J., 1957. MR 0087148