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

General formulation

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

${\displaystyle Lu=-\phi (x),x=(x_{1},x_{2},\dots ,x_{n})\in D}$

with boundary conditions:

${\displaystyle lu=g(x),x\in \partial D}$

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

Problem in the extended domain ${\displaystyle \Omega }$ for the new solution ${\displaystyle u_{\epsilon }(x)}$:

${\displaystyle L_{\epsilon }u_{\epsilon }=-\phi ^{\epsilon }(x),x=(x_{1},x_{2},\dots ,x_{n})\in \Omega }$

${\displaystyle l_{\epsilon }u_{\epsilon }=g^{\epsilon }(x),x\in \partial \Omega }$

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

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

Simple example, 1-dimensional problem

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

${\displaystyle u(0)=0,u(1)=0}$

Prolongation by leading coefficients

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

${\displaystyle {\frac {d}{dx}}k^{\epsilon }(x){\frac {du_{\epsilon }}{dx}}=-\phi ^{\epsilon }(x),0<x<2\quad (2)}$

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

${\displaystyle k^{\epsilon }(x)={\begin{cases}1,&0<x<1\\{\frac {1}{\epsilon ^{2}}},&1<x<2\end{cases}}}$

${\displaystyle \phi ^{\epsilon }(x)={\begin{cases}2,&0<x<1\\2c_{0},&1<x<2\end{cases}}\quad (3)}$

Boundary conditions:

${\displaystyle u_{\epsilon }(0)=0,u_{\epsilon }(2)=0}$

Connection conditions in the point ${\displaystyle x=1}$:

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

where ${\displaystyle [\cdot ]}$ means:

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

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

${\displaystyle u(x)-u_{\epsilon }(x)=O(\epsilon ^{2}),\quad 0<x<1}$

Prolongation by lower-order coefficients

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

${\displaystyle {\frac {d^{2}u_{\epsilon }}{dx^{2}}}-c^{\epsilon }(x)u_{\epsilon }=-\phi ^{\epsilon }(x),\quad 0<x<2\quad (4)}$

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

${\displaystyle c^{\epsilon }(x)={\begin{cases}0,&0<x<1\\{\frac {1}{\epsilon ^{2}}},&1<x<2.\end{cases}}}$

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

Connection conditions in the point ${\displaystyle x=1}$:

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

Error:

${\displaystyle u(x)-u_{\epsilon }(x)=O(\epsilon ),\quad 0<x<1}$

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