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