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 .
YouTube Encyclopedic
-
1/5
Views:811
38 000
11 688
2 036
12 095
-
nanoHUB-U Nanophotonic Modeling L3.02: Finite Difference Time Domain Method
-
Disproving implications with Counterexamples
-
7:3 Boundary Element Methods (Indirect, Potential flow)
-
All you need to know from finite element theory | Part 1 | approximation using basis functions
-
7:3 Boundary Element Methods - Indirect, direct, coupled FEM/BEM
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
This page was last edited on 8 March 2024, at 01:04