# Analysis and Numerics of Some Fractal Boundary Value Problems

Bollettino dell'Unione Matematica Italiana (2013)

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

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topMosco, 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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