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 formula_1, by substituting a given problem
posed on a domain formula_1, with a new problem posed on a simple domain formula_3 containing formula_1.
General formulation.
Assume in some area formula_5 we want to find solution formula_6 of the equation:
with boundary conditions:
The basic idea of fictitious domains method is to substitute a given problem
posed on a domain formula_1, with a new problem posed on a simple shaped domain formula_3 containing formula_1 (formula_12). For example, we can choose "n"-dimensional parallelotope as formula_3.
Problem in the extended domain formula_3 for the new solution formula_15:
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.
formula_21 solution of problem:
Discontinuous coefficient formula_23 and right part of equation previous equation we obtain from expressions:
Boundary conditions:
Connection conditions in the point formula_27:
where formula_29 means:
Equation (1) has analytical solution therefore we can easily obtain error:
Prolongation by lower-order coefficients.
formula_21 solution of problem:
Where formula_34 we take the same as in (3), and expression for formula_35
Boundary conditions for equation (4) same as for (2).
Connection conditions in the point formula_27:
Error: