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 D {displaystyle D} , by substituting a given problemposed on a domain D {displaystyle D} , with a new problem posed on a simple domain Ω {displaystyle Omega } containing D {displaystyle D} . In mathematics, the Fictitious domain method is a method to find the solution of a partial differential equations on a complicated domain D {displaystyle D} , by substituting a given problemposed on a domain D {displaystyle D} , with a new problem posed on a simple domain Ω {displaystyle Omega } containing D {displaystyle D} . Assume in some area D ⊂ R n {displaystyle Dsubset mathbb {R} ^{n}} we want to find solution u ( x ) {displaystyle u(x)} of the equation: with boundary conditions: The basic idea of fictitious domains method is to substitute a given problemposed on a domain D {displaystyle D} , with a new problem posed on a simple shaped domain Ω {displaystyle Omega } containing D {displaystyle D} ( D ⊂ Ω {displaystyle Dsubset Omega } ). For example, we can choose n-dimensional parallelotope as Ω {displaystyle Omega } . Problem in the extended domain Ω {displaystyle Omega } for the new solution u ϵ ( x ) {displaystyle u_{epsilon }(x)} :