Fictitious domain method - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Fictitious domain method is a method to find the solution of a partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s 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

Equation

An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...

with boundary conditions

Boundary value problem

In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the 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 parallelepiped 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

Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...

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

Boundary value problem

In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the 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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