# 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

    ${\frac {d^{2}u}{dx^{2}}}=-2,\quad 0<x<1\quad (1)$
    $u(0)=0,u(1)=0$

### 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