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 $D \subset \mathbb^n$ 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 parallelotope as $\Omega$.

Problem in the extended domain $\Omega$ for the new solution $u_(x)$:

: $L_\epsilon u_\epsilon = - \phi^\epsilon(x), x = (x_1, x_2, \dots , x_n) \in \Omega$

: $l_\epsilon u_\epsilon = g^\epsilon(x), x \in \partial \Omega$

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

:
u_\epsilon (x) \xrightarrow epsilon \rightarrow 0u(x), x \in D

Simple example, 1-dimensional problem

:$\frac = -2, \quad 0 < x < 1 \quad (1)$

:$u(0) = 0, u(1) = 0$

Prolongation by leading coefficients

$u_\epsilon(x)$ solution of problem:

: $\frack^\epsilon(x)\frac = - \phi^(x), 0 < x < 2 \quad (2)$
Discontinuous coefficient $k^(x)$ and right part of equation previous equation we obtain from expressions:

: $k^\epsilon (x)=\begin 1, & 0 < x < 1 \\ \frac, & 1 < x < 2 \end$
: $\phi^\epsilon (x)=\begin 2, & 0 < x < 1 \\ 2c_0, & 1 < x < 2 \end\quad (3)$

Boundary conditions:

: $u_\epsilon(0) = 0, u_\epsilon(2) = 0$

Connection conditions in the point $x = 1$:

:
_\epsilon= 0,\ \left ^\epsilon(x)\frac\right= 0

where \cdot /math> means:

: $$
(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:

: $\frac - c^\epsilon(x)u_\epsilon = - \phi^\epsilon(x), \quad 0 < x < 2 \quad (4)$

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

: $c^\epsilon(x)=\begin 0, & 0 < x < 1 \\ \frac, & 1 < x < 2. \end$

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

Connection conditions in the point $x = 1$:

:
_\epsilon(0)= 0,\ \left frac\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

{{DEFAULTSORT:Fictitious Domain Method
Domain decomposition methods
Applied mathematics