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^{2}u}{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^{2}u_{\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

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