User:Greegg0/sandbox

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In [[mathematics]], the '''fictitious domain method''' is a method to find the solution of a [[partial differential equation]]s on a complicated [[Domain of a function|domain]] <math>D</math>, by substituting a given problem

The '''apple''' is a fruit.

posed on a domain <math>D</math>, with a new problem posed on a simple domain <math>\Omega</math> containing <math>D</math>.

==General formulation==

Assume in some area <math>D \subset \mathbb{R}^n </math> we want to find solution <math>u(x)</math> of the [[equation]]:

: <math>

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

</math>

with [[Boundary value problem|boundary conditions]]:

: <math>

lu = g(x), x \in \partial D

</math>

The basic idea of fictitious domains method is to substitute a given problem

posed on a domain <math>D</math>, with a new problem posed on a simple [[shaped domain]] <math>\Omega</math> containing <math>D</math> (<math>D \subset \Omega</math>). For example, we can choose ''n''-dimensional parallelotope as <math>\Omega</math>.

Problem in the [[extended domain]] <math>\Omega</math> for the new solution <math>u_{\epsilon}(x)</math>:

: <math>

L_\epsilon u_\epsilon = - \phi^\epsilon(x), x = (x_1, x_2, \dots , x_n) \in \Omega

</math>

: <math>

l_\epsilon u_\epsilon = g^\epsilon(x), x \in \partial \Omega

</math>

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

: <math>

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

</math>

== Simple example, 1-dimensional problem ==

:<math>

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

</math>

:<math>

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

</math>

=== Prolongation by leading coefficients ===

<math>u_\epsilon(x)</math> solution of problem:

: <math>

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

</math>

Discontinuous [[coefficient]] <math>k^{\epsilon}(x)</math> and right part of equation previous equation we obtain from expressions:

: <math>

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

\end{cases}

</math>

: <math>

\phi^\epsilon (x)=\begin{cases} 2, & 0 < x < 1 \\ 2c_0, & 1 < x < 2

\end{cases}\quad (3)

</math>

Boundary conditions:

: <math>

u_\epsilon(0) = 0, u_\epsilon(2) = 0

</math>

Connection conditions in the point <math>x = 1</math>:

: <math>

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

</math>

where <math>[ \cdot ]</math> means:

: <math>

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

</math>

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

: <math>

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

</math>

===Prolongation by lower-order coefficients===

<math>u_\epsilon(x)</math> solution of problem:

: <math>

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

</math>

Where <math>\phi^{\epsilon}(x)</math> we take the same as in (3), and expression for <math>c^{\epsilon}(x)</math>

: <math>

c^\epsilon(x)=\begin{cases}

0, & 0 < x < 1 \\

\frac{1}{\epsilon^2}, & 1 < x < 2.

\end{cases}

</math>

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

Connection conditions in the point <math>x = 1</math>:

: <math>

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

</math>

Error:

: <math>

u(x) - u_\epsilon(x) = O(\epsilon), \quad 0 < x < 1

</math>

==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.&nbsp;79–90

{{Numerical PDE}}