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 parallelotope 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}\quad (3) $$

Boundary conditions:



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

Connection conditions in the point $$x = 1$$:



[u_\epsilon] = 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} 0,                   &  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