Immersed interface methods. Final report

Cartesian grid methods encompass a wide variety of techniques used to solve partial differential equations in more than one space dimension on uniform Cartesian grids even when the underlying geometry is complex and not aligned with the grid. The authors` groups work on Immersed Interface Methods (IIM) was originally motivated by the desire to understand and improve the ``Immersed Boundary Method``, developed by Charles Peskin to solve incompressible Navier-Stokes equations in complicated geometries with moving elastic boundaries. This report briefly discusses the development of the Immersed Interface Methods and gives examples of application of the method in solving several partial differential equations