Mosco, Umberto:
Analysis and Numerics of Some Fractal Boundary Value Problems
Bollettino dell'Unione Matematica Italiana Serie 9 6 (2013), fasc. n.1, p. 53-73, (English)
pdf (353 Kb), djvu (222 Kb). | MR 3077113 | Zbl 1280.35035
Sunto
We describe some recent results for boundary value problems with fractal boundaries. Our aim is to show that the numerical approach to boundary value problems, so much cherished and in many ways pioneering developed by Enrico Magenes, takes on a special relevance in the theory of boundary value problems in fractal domains and with fractal operators. In this theory, in fact, the discrete numerical analysis of the problem precedes the, and indeed give rise to, the asymptotic continuous problem, reverting in a sense the process consisting in deriving discrete approximations from the PDE itself by finite differences or finite elements. As an illustration of this point, in this note we describe some recent results on: the approximation of a fractal Laplacian by singular elliptic partial differential operators, by Vivaldi and the author; the asymptotic of degenerate Laplace equations in domains with a fractal boundary, by Capitanelli-Vivaldi; the fast heat conduction on a Koch interface, by Lancia-Vernole and co-authors. We point out that this paper has an illustrative purpose only and does not aim at providing a survey on the subject.
Referenze Bibliografiche
[1]
Y. ACHDOU -
C. SABOT -
N. TCHOU,
A multiscale numerical method for Poisson problems in some ramified domains with fractal boundary,
Multiscale Model. Simul. 5 (3) (
2006), 828-860. |
fulltext (doi) |
MR 2257237 |
Zbl 1149.35023[2]
H. ATTOUCH,
Variational Convergence for Functions and Operators,
Pitman Advanced Publishing Program, London
1984. |
MR 773850 |
Zbl 0561.49012[5]
J. R. CANNON -
G. H. MEYER,
On a diffusion in a fractured medium,
SIAM J. Appl. Math.,
3 (
1971), 434-448. |
Zbl 0266.35002[7]
R. CAPITANELLI -
M. A. VIVALDI,
On the Laplacean transfer across a fractal mixture,
Asymptotic Analysis, to appear. DOI 10.3233/ASY-2012-1149. |
MR 3100114[8]
M. CEFALO -
G. DELLA'ACQUA -
M.R. LANCIA,
Numerical approximation of transmission problems across Koch-type highly conductive layers,
Appl. Math. Comp.,
218 (
2012), 5453-5473. |
fulltext (doi) |
MR 2870066 |
Zbl 06045310[9]
M. CEFALO -
M.R. LANCIA -
H. LIANG,
Heat flow problems across fractal mixtures: regularity results of the solutions and numerical approximation,
Differential and Integral Equations, in print. |
MR 3100075 |
Zbl 1299.65228[10] E. EVANS, Extension Operators and Finite Elements for Fractal Boundary value Problems, PhD thesis, Worcester Polytecnic Institute, USA, 2011.
[11]
P. GRISVARD,
Elliptic problems in nonsmooth domains, Advanced Publishing Program, London
1985. |
MR 775683 |
Zbl 0695.35060[14]
A. JONSSON -
H. WALLIN,
Function Spaces on Subsets of $\mathbb{R}^n$, Part 1 (
Math. Reports) Vol.
2. London:
Harwood Acad. Publ. 1984. |
MR 820626 |
Zbl 0875.46003[15]
K. KUWAE -
T. SHIOYA,
Convergence of spectral structures: a functional analytic theory and its applications to spectral geometry,
Comm. in Anal. and Geom.,
11 (4) (
2003), 599-673. |
fulltext (doi) |
MR 2015170 |
Zbl 1092.53026[17]
M. R. LANCIA -
E. VACCA,
Numerical approximation of heat flow problems across highly conductive layers, ``
Quaderni di Matematica'',
Dept. Me. Mo. Mat., U. Roma La Sapienza,
2007. |
MR 2522082[18]
M. R. LANCIA -
M. A. VIVALDI,
Asymptotic convergence for energy forms,
Adv. Math. Sc. Appl.,
13 (
2003), 315-341. |
MR 2002727 |
Zbl 1205.35129[19]
M. R. LANCIA -
P. VERNOLE,
Convergence results for parabolic transmission problems across highly conductive layers with small capacity,
AMSA,
16 (
2006), 411-445. |
MR 2356123 |
Zbl 1137.35042[21]
M. R. LANCIA -
P. VERNOLE,
Semilinear evolution transmission problems across fractal layers,
Nonlinear Anal. Th. and Appl.,
75 (
2012), 4222-4240. |
fulltext (doi) |
MR 2921985 |
Zbl 1241.31012[22]
M. R. LANCIA -
P. VERNOLE,
Semilinear fractal problems: approximation and regularity results,
Nonlinear Anal. Th. and Appl., to appear. DOI: 10.1016/j.na.2012.08.020. |
fulltext (doi) |
MR 3010768[23] H. LIANG, On the construction of certain fractal mixtures, MS Thesis, WPI, 2009.
[25]
U. MOSCO,
Variational fractals,
Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4) Vol.
XXV (
1997), 683-712. |
fulltext EuDML |
MR 1655537[26]
U. MOSCO,
Harnack inequalities on scale irregular Sierpiński gaskets,
Nonlinear Problems in Math. Physics and Related Topics II, Edited by
Birman et al.,
Kluwer Acad./Plenum Publ. (New York,
2002), 305-328. |
fulltext (doi) |
MR 1972003 |
Zbl 1030.28004[27] U.MOSCO -
M. A. VIVALDI,
An example of fractal singular homogenization,
Georgian Math. J. 14 (1) (
2007), 169-194. |
MR 2323380 |
Zbl 1135.35015[30]
U. MOSCO -
M. A. VIVALDI,
Thin fractal fibers,
Math. Methods in the Applied Sciences, to appear. DOI: 10.1002/mma.1621. |
MR 3108826[31]
H. PHAM HUY -
E. SANCHEZ PALENCIA,
Phénomènes des transmission à travers des couches minces de conductivité élevée,
J. Math Anal. Appl.,
47 (
1974), 284-309. |
fulltext (doi) |
MR 400916 |
Zbl 0286.35007[32] E. VACCA, Galerkin Approximation for Highly Conductive Layers, PhD Thesis, Dept. Me. Mo. Mat., U. Roma La Sapienza, 2005.
[33] R. WASYK, Numerical solution of a transmission problem with pre fractal interface, PhD Thesis, WPI, 2007.