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 [math]\displaystyle{ D }[/math], by substituting a given problem posed on a domain [math]\displaystyle{ D }[/math], with a new problem posed on a simple domain [math]\displaystyle{ \Omega }[/math] containing [math]\displaystyle{ D }[/math].

General formulation

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

[math]\displaystyle{ Lu = - \phi(x), x = (x_1, x_2, \dots , x_n) \in D }[/math]

with boundary conditions:

[math]\displaystyle{ lu = g(x), x \in \partial D }[/math]

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

Problem in the extended domain [math]\displaystyle{ \Omega }[/math] for the new solution [math]\displaystyle{ u_{\epsilon}(x) }[/math]:

[math]\displaystyle{ L_\epsilon u_\epsilon = - \phi^\epsilon(x), x = (x_1, x_2, \dots , x_n) \in \Omega }[/math]

[math]\displaystyle{ l_\epsilon u_\epsilon = g^\epsilon(x), x \in \partial \Omega }[/math]

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

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

Simple example, 1-dimensional problem

[math]\displaystyle{ \frac{d^2u}{dx^2} = -2, \quad 0 \lt x \lt 1 \quad (1) }[/math]

[math]\displaystyle{ u(0) = 0, u(1) = 0 }[/math]

Prolongation by leading coefficients

[math]\displaystyle{ u_\epsilon(x) }[/math] solution of problem:

[math]\displaystyle{ \frac{d}{dx}k^\epsilon(x)\frac{du_\epsilon}{dx} = - \phi^{\epsilon}(x), 0 \lt x \lt 2 \quad (2) }[/math]

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

[math]\displaystyle{ k^\epsilon (x)=\begin{cases} 1, & 0 \lt x \lt 1 \\ \frac{1}{\epsilon^2}, & 1 \lt x \lt 2 \end{cases} }[/math]

[math]\displaystyle{ \phi^\epsilon (x)=\begin{cases} 2, & 0 \lt x \lt 1 \\ 2c_0, & 1 \lt x \lt 2 \end{cases}\quad (3) }[/math]

Boundary conditions:

[math]\displaystyle{ u_\epsilon(0) = 0, u_\epsilon(2) = 0 }[/math]

Connection conditions in the point [math]\displaystyle{ x = 1 }[/math]:

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

where [math]\displaystyle{ [ \cdot ] }[/math] means:

[math]\displaystyle{ [p(x)] = p(x + 0) - p(x - 0) }[/math]

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

[math]\displaystyle{ u(x) - u_\epsilon(x) = O(\epsilon^2), \quad 0 \lt x \lt 1 }[/math]

Prolongation by lower-order coefficients

[math]\displaystyle{ u_\epsilon(x) }[/math] solution of problem:

[math]\displaystyle{ \frac{d^2u_\epsilon}{dx^2} - c^\epsilon(x)u_\epsilon = - \phi^\epsilon(x), \quad 0 \lt x \lt 2 \quad (4) }[/math]

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

[math]\displaystyle{ c^\epsilon(x)=\begin{cases} 0, & 0 \lt x \lt 1 \\ \frac{1}{\epsilon^2}, & 1 \lt x \lt 2. \end{cases} }[/math]

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

Connection conditions in the point [math]\displaystyle{ x = 1 }[/math]:

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

Error:

[math]\displaystyle{ u(x) - u_\epsilon(x) = O(\epsilon), \quad 0 \lt x \lt 1 }[/math]

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