World Library  
Flag as Inappropriate
Email this Article

Fictitious domain method

Article Id: WHEBN0030334381
Reproduction Date:

Title: Fictitious domain method  
Author: World Heritage Encyclopedia
Language: English
Subject: Domain decomposition methods, Neumann–Neumann methods, Abstract additive Schwarz method, Neumann–Dirichlet method, Mortar methods
Publisher: World Heritage Encyclopedia

Fictitious domain method

In mathematics, the Fictitious domain method is a method to find the solution of a partial differential equations on a complicated domain D, by substituting a given problem posed on a domain D, with a new problem posed on a simple domain \Omega containing D.

General formulation

Assume in some area D \subset \mathbb{R}^n we want to find solution u(x) of the equation:

Lu = - \phi(x), x = (x_1, x_2, \dots , x_n) \in D

with boundary conditions:

lu = g(x), x \in \partial D \,

The basic idea of fictitious domains method is to substitute a given problem posed on a domain D, with a new problem posed on a simple shaped domain \Omega containing D (D \subset \Omega). For example, we can choose n-dimensional parallelepiped as \Omega.

Problem in the extended domain \Omega for the new solution u_{\epsilon}(x):

L_\epsilon u_\epsilon = - \phi^\epsilon(x), x = (x_1, x_2, \dots , x_n) \in \Omega
l_\epsilon u_\epsilon = g^\epsilon(x), x \in \partial \Omega

It is necessary to pose the problem in the extended area so that the following condition is fulfilled:

u_\epsilon (x) \xrightarrow[\epsilon \rightarrow 0]{ } u(x), x \in D \,

Simple example, 1-dimensional problem

\frac{d^2u}{dx^2} = -2, \quad 0 < x < 1 \quad (1)
u(0) = 0, u(1) = 0 \,

Prolongation by leading coefficients

u_\epsilon(x) solution of problem:

\frac{d}{dx}k^\epsilon(x)\frac{du_\epsilon}{dx} = - \phi^{\epsilon}(x), 0 < x < 2 \quad (2)

Discontinuous coefficient k^{\epsilon}(x) and right part of equation previous equation we obtain from expressions:

k^\epsilon (x)=\begin{cases} 1, & 0 < x < 1 \\ \frac{1}{\epsilon^2}, & 1 < x < 2 \end{cases}
\phi^\epsilon (x)=\begin{cases} 2, & 0 < x < 1 \\ 2c_0, & 1 < x < 2 \end{cases}

Boundary conditions:

u_\epsilon(0) = 0, u_\epsilon(1) = 0

Connection conditions in the point x = 1:

[u_\epsilon(0)] = 0,\ \left[k^\epsilon(x)\frac{du_\epsilon}{dx}\right] = 0

where [ \cdot ] means:

[p(x)] = p(x + 0) - p(x - 0) \,

Equation (1) has analytical solution therefore we can easily obtain error:

u(x) - u_\epsilon(x) = O(\epsilon^2), \quad 0 < x < 1

Prolongation by lower-order coefficients

u_\epsilon(x) solution of problem:

\frac{d^2u_\epsilon}{dx^2} - c^\epsilon(x)u_\epsilon = - \phi^\epsilon(x), \quad 0 < x < 2 \quad (4)

Where \phi^{\epsilon}(x) we take the same as in (3), and expression for c^{\epsilon}(x)

c^\epsilon(x)=\begin{cases} 1, & 0 < x < 1 \\ \frac{1}{\epsilon^2}, & 1 < x < 2 \end{cases}

Boundary conditions for equation (4) same as for (2).

Connection conditions in the point x = 1:

[u_\epsilon(0)] = 0,\ \left[\frac{du_\epsilon}{dx}\right] = 0


u(x) - u_\epsilon(x) = O(\epsilon), \quad 0 < x < 1


  • P.N. Vabishchevich, The Method of Fictitious Domains in Problems of Mathematical Physics, Izdatelstvo Moskovskogo Universiteta, Moskva, 1991.
  • Smagulov S. Fictitious Domain Method for Navier–Stokes equation, Preprint CC SA USSR, 68, 1979.
  • Bugrov A.N., Smagulov S. Fictitious Domain Method for Navier–Stokes equation, Mathematical model of fluid flow, Novosibirsk, 1978, p. 79–90
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.