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

, by substituting a given problem posed on a domain

, with a new problem posed on a simple domain

Assume in some area

of the equation: with boundary conditions: 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

For example, we can choose n-dimensional parallelotope as

Problem in the extended domain

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

solution of problem: Discontinuous coefficient

and right part of equation previous equation we obtain from expressions: Boundary conditions: Connection conditions in the point

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

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

Connection conditions in the point