Analysis and Numerics of Some Fractal Boundary Value Problems

Umberto Mosco

Bollettino dell'Unione Matematica Italiana (2013)

  • Volume: 6, Issue: 1, page 53-73
  • ISSN: 0392-4041

Abstract

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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.

How to cite

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Mosco, Umberto. "Analysis and Numerics of Some Fractal Boundary Value Problems." Bollettino dell'Unione Matematica Italiana 6.1 (2013): 53-73. <http://eudml.org/doc/294012>.

@article{Mosco2013,
abstract = {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.},
author = {Mosco, Umberto},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {53-73},
publisher = {Unione Matematica Italiana},
title = {Analysis and Numerics of Some Fractal Boundary Value Problems},
url = {http://eudml.org/doc/294012},
volume = {6},
year = {2013},
}

TY - JOUR
AU - Mosco, Umberto
TI - Analysis and Numerics of Some Fractal Boundary Value Problems
JO - Bollettino dell'Unione Matematica Italiana
DA - 2013/2//
PB - Unione Matematica Italiana
VL - 6
IS - 1
SP - 53
EP - 73
AB - 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.
LA - eng
UR - http://eudml.org/doc/294012
ER -

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