Fictitious domains with separable preconditioners versus unstructured adapted meshes

Projet MENUSINFictitious domain approach is a technique to solve partial differential equations on arbitrary domains, with an iterative method which is preconditioned by the separable linear operator attached to the uniform grid closest to . It is very fast but it uses grids which are not very smooth near the boundaries. In order to evaluate these methods and compare them with more classical methods like the finite element method on unstructured meshes and the preconditionned conjugate gradient method, we have applied two of these methods to an industrial problem for which the solution is known : the computation of the pressure distribution of an airfoil with lift in an irrotational flow. At equal number of points the fictitious domain approach is much faster. At equal precision the fictitious domain approach is also faster, however if a mesh optimizer is used the finite element method has similar execution times