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 .
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 containing (). For example, we can choose n-dimensional parallelotope as .
Problem in the for the new solution :
It is necessary to pose the problem in the extended area so that the following condition is fulfilled:
Simple example, 1-dimensional problem[]
Prolongation by leading coefficients[]
solution of problem:
Discontinuous coefficient and right part of equation previous equation we obtain from expressions:
Boundary conditions:
Connection conditions in the point :
where means:
Equation (1) has analytical solution therefore we can easily obtain error:
Prolongation by lower-order coefficients[]
solution of problem:
Where we take the same as in (3), and expression for
Boundary conditions for equation (4) same as for (2).
Connection conditions in the point :
Error:
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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Numerical methods for partial differential equations