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 .
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:
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:
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:
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 .
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:
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:
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: