Adaptivity in fluid-structure systems

Abstract: In this work an efficient computational tool is presented to calculate the modes and frequencies of resonance for a fluid-structure system. The solution by the finite element methods of such problems produces an error that should be analyzed. It is necessary to be able to detect automatically the areas in which the error is big and refine the mesh to obtain a better approximation. The definition of appropriate a posteriori error estimators becomes the main problem, and is necessary to obtain mathematical proof to exhibit the good behavior of them. The error estimators for three different eigenvalue problems are presented, and they are equivalent to the error up to higher order terms. The problems that we will consider are the vibrations of: 1) an elastic structure; 2) the fluid inside a rigid cavity; and 3) an elastic structure with fluids in their interior. For the fluid we chose the displacement field as unknown, and the Raviart-Thomas elements are used, since they don't produces spurious eigenvalues and they have good optimal order of convergence.