Stochastic methods for Dirichlet problems
type: Item
instance of: scholarly article
title: Stochastic methods for Dirichlet problems
zbMATH Open document ID: 1084.65011
author: R. Ardanuy-Albajar
author: B. A. Wade
author: Jesús Vigo-Aguiar
published in: JMMA. Journal of Mathematical Modelling and Algorithms
publication date: 2006-01-23T00:00:00Z
review text: It is well-known that solutions of second order Dirichlet problems possess probabilistic representations in terms of Ito type stochastic differential equations (SDEs). The authors present a numerical solution method for Dirichlet boundary value problems based on those probabilistic representations. Note that, with this idea, it is possible to find solution values of a partial differential equation at isolated points without having to construct any kind of space-mesh and without knowing approximations for the solution at any other points. Their method is similar to the approach of \textit{F. M. Buchmann} and \textit{W. P. Petersen} [BIT 43, No.~3, 519--540 (2003; Zbl 1050.65001)]. However, their approach differs primarily in the handling of the boundary. Some results of numerical examples are presented, especially for applying these techniques to Laplace and Poisson equations on the unit disk. Euler- and trapezoidal-type methods are used to approximate related SDEs and involved functionals.
reviewed by: Henri Schurz
Mathematics Subject Classification ID: 65C30
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 37H10
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M20
zbMATH DE Number: 5000193
zbMATH Keywords: stochastic-numerical methods
zbMATH Keywords: elliptic problems
zbMATH Keywords: second order Dirichlet problem
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: Laplace and Poisson equations
zbMATH Keywords: maximum principle
zbMATH Keywords: probabilistic representation
zbMATH Keywords: Euler method
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
full work available at URL: s10852-005-9007-0
OpenAlex ID: W1970794569
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DOI: 10.1007/S10852-005-9007-0
description: scientific article; zbMATH DE number 5000193