# 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 $u(0)=0,u(1)=0$ ### Prolongation by leading coefficients

$u_{\epsilon }(x)$ solution of problem:

${\frac {d}{dx}}k^{\epsilon }(x){\frac {du_{\epsilon }}{dx}}=-\phi ^{\epsilon }(x),0 Discontinuous coefficient $k^{\epsilon }(x)$ and right part of equation previous equation we obtain from expressions:

$k^{\epsilon }(x)={\begin{cases}1,&0 $\phi ^{\epsilon }(x)={\begin{cases}2,&0 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 ### 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 Where $\phi ^{\epsilon }(x)$ we take the same as in (3), and expression for $c^{\epsilon }(x)$ $c^{\epsilon }(x)={\begin{cases}0,&0 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 ## 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