# 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
containing
.

## General formulation

Assume in some area we want to find solution 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
containing
(
). For example, we can choose *n*-dimensional parallelotope as
.

Problem in the extended domain 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