A Fokker-Planck type approximation of parabolic PDEs with oblique boundary data
type: Item
instance of: scholarly article
title: A Fokker-Planck type approximation of parabolic PDEs with oblique boundary data
zbMATH Open document ID: 1372.35158
author: Damon Alexander
author: Inwon Kim
published in: Transactions of the American Mathematical Society
publication date: 2016-03-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1403.2778
Mathematics Subject Classification ID: 35K59
Mathematics Subject Classification ID: 35K10
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35K61
Mathematics Subject Classification ID: 35Q84
Mathematics Subject Classification ID: 65M06
zbMATH DE Number: 6551581
zbMATH Keywords: Fokker-Planck equation
zbMATH Keywords: quasilinear parabolic partial differential equaiton
zbMATH Keywords: approximation by extension
zbMATH Keywords: boundary data
MaRDI profile type: Publication
OpenAlex ID: W1616108595
arXiv ID: 1403.2778
cites work: On Neumann and oblique derivatives boundary conditions for nonlocal elliptic equations
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cites work: Parabolic equations with measurable coefficients
cites work: Q4125241
cites work: Nonlinear Oblique Boundary Value Problems for Nonlinear Elliptic Equations
cites work: Stochastic differential equations with reflecting boundary conditions
DOI: 10.1090/TRAN/6521
review text: The authors consider solutions to quasilinear parabolic partial differential equations with zero oblique boundary data in a bounded domain. They approximate its solutions by solutions to a Fokker-Planck type equation in the whole space. The equation corresponds to the quasilinear one inside of the domain and a Fokker-Planck equation with a penalizing drift term outside of the domain. The penalization parameter \(N\) is the strength of the drift term. The convergence is locally uniform as \(N\to \infty\) and error estimates in the penalization parameter \(N\) are obtained.NEWLINENEWLINEThe result is first derived for the linear case, which is a Fokker-Planck equation with Neumann boundary data but not necessarily constant diffusion matrix. Here the construction of a supersolution for the approximate problem is based on the solution of the limit problem inside of the domain as well as an extension using the explicit form of the drift term. The general case is dealt with by a second approximation step, which interpolates between the linear second-order operator with diffusion matrix matching the boundary data outside of the domain and the quasilinear operator inside of the domain.NEWLINENEWLINEThe analysis is complemented by a numerical investigation, in which the error bound is numerically verified.
reviewed by: André Schlichting
description: scientific article; zbMATH DE number 6551581