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In mathematics, the Fictitious domain method is a method to find the solution of a partial differential equations on a complicated domain
, by substituting a given problem
posed on a domain
, with a new problem posed on a simple domain
containing
.
General formulation
Assume in some area
we want to find solution
of the equation:
![Lu=-\phi (x),x=(x_{1},x_{2},\dots ,x_{n})\in D](https://wikimedia.org/api/rest_v1/media/math/render/svg/07c0b0ba5c10a40e57323474d6386cb2f1fc2d43)
with boundary conditions:
![lu=g(x),x\in \partial D\,](https://wikimedia.org/api/rest_v1/media/math/render/svg/88e930ae9f4fdbd2b433c40e6075f31db62267b1)
The basic idea of fictitious domains method is to substitute a given problem
posed on a domain
, with a new problem posed on a simple shaped domain
containing
(
). For example, we can choose n-dimensional parallelepiped as
.
Problem in the extended domain
for the new solution
:
![L_{\epsilon }u_{\epsilon }=-\phi ^{\epsilon }(x),x=(x_{1},x_{2},\dots ,x_{n})\in \Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/42ec13be8a681f4f162b6570ff737db585c74513)
![l_{\epsilon }u_{\epsilon }=g^{\epsilon }(x),x\in \partial \Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b0e7f6d8b50816b1fcdee0f3ea7076cc8e20216)
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\,](https://wikimedia.org/api/rest_v1/media/math/render/svg/606043cef65dd8d371a3c99cb703693f0274d4c8)
Simple example, 1-dimensional problem
![{\frac {d^{2}u}{dx^{2}}}=-2,\quad 0<x<1\quad (1)](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccb24852c24bd282962f5c26bd8ac11b01146082)
![u(0)=0,u(1)=0\,](https://wikimedia.org/api/rest_v1/media/math/render/svg/58e001b1683e7a7400fe431e0159869149b4ae6b)
Prolongation by leading coefficients
solution of problem:
![{\frac {d}{dx}}k^{\epsilon }(x){\frac {du_{\epsilon }}{dx}}=-\phi ^{{\epsilon }}(x),0<x<2\quad (2)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d93923f90e90cb91fc4004c0552a6fb35e147e36)
Discontinuous coefficient
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98e456ef06f07163ef06f31734a860ba71133453)
![(3)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ff45737214013a8e04d59d0de54318086be26a)
![\phi ^{\epsilon }(x)={\begin{cases}2,&0<x<1\\2c_{0},&1<x<2\end{cases}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43ed748f190f3fc4302de5af5a175f49f0bf9570)
Boundary conditions:
![u_{\epsilon }(0)=0,u_{\epsilon }(1)=0](https://wikimedia.org/api/rest_v1/media/math/render/svg/2dff1b23f61d31def165f2a2ed1a37d3e004e88d)
Connection conditions in the point
:
![[u_{\epsilon }(0)]=0,\ \left[k^{\epsilon }(x){\frac {du_{\epsilon }}{dx}}\right]=0](https://wikimedia.org/api/rest_v1/media/math/render/svg/e63ad30d525601e7c1ca6247ded0c299aea46d1f)
where
means:
![[p(x)]=p(x+0)-p(x-0)\,](https://wikimedia.org/api/rest_v1/media/math/render/svg/881dc4c20d5f3da7dc9179eea4ce569bcd1f3408)
Equation (1) has analytical solution therefore we can easily obtain error:
![u(x)-u_{\epsilon }(x)=O(\epsilon ^{2}),\quad 0<x<1](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd1753b6d7b35288a87ccb413d50dd8fe0c666cc)
Prolongation by lower-order coefficients
solution of problem:
![{\frac {d^{2}u_{\epsilon }}{dx^{2}}}-c^{\epsilon }(x)u_{\epsilon }=-\phi ^{\epsilon }(x),\quad 0<x<2\quad (4)](https://wikimedia.org/api/rest_v1/media/math/render/svg/530ca2fd47e3e6c61194a95eb1b0f9b2bff81c91)
Where
we take the same as in (3), and expression for
![c^{\epsilon }(x)={\begin{cases}1,&0<x<1\\{\frac {1}{\epsilon ^{2}}},&1<x<2\end{cases}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23ca55c69f123e4533a4ed38c4e3fd628896e836)
Boundary conditions for equation (4) same as for (2).
Connection conditions in the point
:
![[u_{\epsilon }(0)]=0,\ \left[{\frac {du_{\epsilon }}{dx}}\right]=0](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdf31b2d4d05429863841e47193564a63b8e4a9f)
Error:
![u(x)-u_{\epsilon }(x)=O(\epsilon ),\quad 0<x<1](https://wikimedia.org/api/rest_v1/media/math/render/svg/415fd0de098ebba569f61848e516ad1aac209d82)
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
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