Method of Green's potentials for elliptic PDEs in domains with random apertures
| Autor: | Yuri A. Melnikov, Viktor Reshniak |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Discretization
Monte Carlo method Dimension (graph theory) 01 natural sciences Domain (mathematical analysis) Theoretical Computer Science Green S 65N38 65N80 65N85 65C05 chemistry.chemical_compound FOS: Mathematics Applied mathematics Mathematics - Numerical Analysis 0101 mathematics Differential (infinitesimal) Mathematics Numerical Analysis Applied Mathematics Probability (math.PR) General Engineering Ranging Numerical Analysis (math.NA) Integral equation 010101 applied mathematics Computational Mathematics Computational Theory and Mathematics chemistry Software Mathematics - Probability |
| Popis: | Problems with topological uncertainties appear in many fields ranging from nano-device engineering to the design of bridges. In many of such problems, a part of the domains boundaries is subjected to random perturbations making inefficient conventional schemes that rely on discretization of the whole domain. In this paper, we study elliptic PDEs in domains with boundaries comprised of a deterministic part and random apertures, and apply the method of modified potentials with Green’s kernels defined on the deterministic part of the domain. This approach allows to reduce the dimension of the original differential problem by reformulating it as a boundary integral equation posed on the random apertures only. The multilevel Monte Carlo method is then applied to this modified integral equation and its optimal $$\epsilon ^{-2}$$ asymptotical complexity is shown. Finally, we provide the qualitative analysis of the proposed technique and support it with numerical results. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|