Spectral representations, and approximations, of divergence-free vector fields
Authors:
Giles Auchmuty and Douglas R. Simpkins
Journal:
Quart. Appl. Math. 74 (2016), 429-441
MSC (2010):
Primary 35Q35, 35P05, 41A99, and, 76M22
DOI:
https://doi.org/10.1090/qam/1427
Published electronically:
June 20, 2016
MathSciNet review:
3518223
Full-text PDF Free Access
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Additional Information
Abstract: Special solutions of the equation for a solenoidal vector field subject to prescribed flux boundary conditions are described. A unique gradient solution is found and proved to be the least energy solution of the problem. This solution has a representation in terms of certain $\Sigma -\mbox {Steklov}-$eigenvalues and eigenfunctions. Error estimates for finite approximations of these solutions are obtained. Some results of computational simulations for two-dimensional and axisymmetrical problems are presented.
References
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- Giles Auchmuty, Spectral characterization of the trace spaces $H^s(\partial \Omega )$, SIAM J. Math. Anal. 38 (2006), no. 3, 894–905. MR 2262947, DOI 10.1137/050626053
- Giles Auchmuty, Bases and comparison results for linear elliptic eigenproblems, J. Math. Anal. Appl. 390 (2012), no. 1, 394–406. MR 2885782, DOI 10.1016/j.jmaa.2012.01.051
- Giles Auchmuty, Finite energy solutions of self-adjoint elliptic mixed boundary value problems, Math. Methods Appl. Sci. 33 (2010), no. 12, 1446–1462. MR 2680685, DOI 10.1002/mma.1258
- F. Hecht, A le Hyaric, K. Ohtsuka and O. Pironneau, “FreeFEM++, v3.29,” 17/3/2014 (online), available at http://www.freefem.org/ff++/ftp/freefem++doc.pdf
- F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3-4, 251–265. MR 3043640, DOI 10.1515/jnum-2012-0013
- Douglas R. Simpkins, Steklov Eigenfunctions; Applications to Div-Curl systems using FreeFEM++, project report. Available at http://www.math.uh.edu/~giles/simpkins
References
- Giovanni P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, Linearized steady problems, Springer Tracts in Natural Philosophy, vol. 38, Springer-Verlag, New York, 1994. MR 1284205 (95i:35216a), DOI 10.1007/978-1-4612-5364-8
- Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology. Vol. 3, Spectral theory and applications, with the collaboration of Michel Artola and Michel Cessenat, translated from the French by John C. Amson, Springer-Verlag, Berlin, 1990. MR 969367 (91h:00004a)
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660 (93f:28001)
- Giles Auchmuty, Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Numer. Funct. Anal. Optim. 25 (2004), no. 3-4, 321–348. MR 2072072 (2005d:35187), DOI 10.1081/NFA-120039655
- Giles Auchmuty, Spectral characterization of the trace spaces $H^s(\partial \Omega )$, SIAM J. Math. Anal. 38 (2006), no. 3, 894–905 (electronic). MR 2262947 (2008b:46054), DOI 10.1137/050626053
- Giles Auchmuty, Bases and comparison results for linear elliptic eigenproblems, J. Math. Anal. Appl. 390 (2012), no. 1, 394–406. MR 2885782, DOI 10.1016/j.jmaa.2012.01.051
- Giles Auchmuty, Finite energy solutions of self-adjoint elliptic mixed boundary value problems, Math. Methods Appl. Sci. 33 (2010), no. 12, 1446–1462. MR 2680685 (2011f:35084), DOI 10.1002/mma.1258
- F. Hecht, A le Hyaric, K. Ohtsuka and O. Pironneau, “FreeFEM++, v3.29,” 17/3/2014 (online), available at http://www.freefem.org/ff++/ftp/freefem++doc.pdf
- F. Hecht, New development in freefem++, J. Numer. Math. 20 (2012), no. 3-4, 251–265. MR 3043640
- Douglas R. Simpkins, Steklov Eigenfunctions; Applications to Div-Curl systems using FreeFEM++, project report. Available at http://www.math.uh.edu/~giles/simpkins
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Additional Information
Giles Auchmuty
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77204-3008
MR Author ID:
28195
Email:
auchmuty@uh.edu
Douglas R. Simpkins
Affiliation:
Weatherford International, Houston, Texas
Email:
SimpkinsDouglas@yahoo.com
Received by editor(s):
December 29, 2014
Published electronically:
June 20, 2016
Additional Notes:
The research of the first author was partially supported by NSF award DMS 11008754
Article copyright:
© Copyright 2016
Brown University
