Structure of liquids and glasses

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 $\mathrm{\Omega }$ containing $D$.

General formulation

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

$Lu=-\varphi (x),x=(x_1,x_2,\dots ,x_n)\in D$

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 $\mathrm{\Omega }$ containing $D$ ($D\subset \mathrm{\Omega }$). For example, we can choose n-dimensional parallelepiped as $\mathrm{\Omega }$.

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

$L_{ϵ}{u}_{ϵ}=-{\varphi }^{ϵ}(x),x=(x_1,x_2,\dots ,x_n)\in \mathrm{\Omega }$

$l_{ϵ}{u}_{ϵ}=g^{ϵ}(x),x\in \partial \mathrm{\Omega }$

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

$u_{ϵ}(x)\underset{ϵ\to 0}{\overset{}{\to }}u(x),x\in D$

Simple example, 1-dimensional problem

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

$u(0)=0,u(1)=0$

Prolongation by leading coefficients

$u_{ϵ}(x)$ solution of problem:

$\frac{d}{dx}{k}^{ϵ}(x)\frac{d{u}_{ϵ}}{dx}=-{\varphi }^{ϵ}(x),0<x<2(2)$

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

$k^{ϵ}(x)=\{\begin{array}{ll}1,& 0<x<1\\ \frac{1}{{ϵ}^2},& 1<x<2\end{array}$

$(3)$

${\varphi }^{ϵ}(x)=\{\begin{array}{ll}2,& 0<x<1\\ 2c_0,& 1<x<2\end{array}$

Boundary conditions:

$u_{ϵ}(0)=0,u_{ϵ}(1)=0$

Connection conditions in the point $x=1$:

$[u_{ϵ}(0)]=0,[k^{ϵ}(x)\frac{d{u}_{ϵ}}{dx}]=0$

where $[\cdot ]$ means:

$[p(x)]=p(x+0)-p(x-0)$

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

$u(x)-u_{ϵ}(x)=O({ϵ}^2),0<x<1$

Prolongation by lower-order coefficients

$u_{ϵ}(x)$ solution of problem:

$\frac{d^2{u}_{ϵ}}{d{x}^2}-c^{ϵ}(x)u_{ϵ}=-{\varphi }^{ϵ}(x),0<x<2(4)$

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

$c^{ϵ}(x)=\{\begin{array}{ll}1,& 0<x<1\\ \frac{1}{{ϵ}^2},& 1<x<2\end{array}$

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

Connection conditions in the point $x=1$:

$[u_{ϵ}(0)]=0,\left[\frac{d{u}_{ϵ}}{dx}\right]=0$

Error:

$u(x)-u_{ϵ}(x)=O(ϵ),0<x<1$

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