A computational setting for the Immersed Boundary Method employing an adaptive mesh refinement is presented. Enhanced accuracy for the method is attained locally by covering an immersed boundary vicinity with a sequence of nested, progressively finer rectangular grid patches which dynamically follow the immersed boundary motion. The set of equations describing the interaction between a non stationary, viscous incompressible fluid and an immersed elastic boundary is solved by coupling a projection method, especially designed for locally refined meshes, to an implicit formulation of the Immersed Boundary Method. The main contributions of this work concern the formulation and the implementation of a multilevel self adaptive version of the Immersed Boundary Method on locally refined meshes. This approach is tested for a particular two-dimensional model problem, for which no significant difference is found between the solutions obtained on a mesh refined locally around the immersed boundary, and on the associated uniform mesh, built with the resolution of the  finest level.
 

Roma, A.M.; Peskin, C.S.; Berger, M.J.: An adaptive version of the Immersed Boundary Method, Journal of Computational Physics, 153, 509-534 (1999).