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Fictitious domain method

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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}
(3)
\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

Error:

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

Literature

  • 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
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