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. 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:  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:  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: