A nonconforming rectangular finite element method is proposed to solve a fluid structure interaction problem characterized by the Darcy-Stokes-Brinkman Equation with discontinuous coefficients across the interface of different structures. A uniformly stable mixed finite element together with Nitsche-type matching conditions that automatically adapt to the coupling of different sub-problem combinations are utilized in the discrete algorithm. Compared with other finite element methods in the literature, the new method has some distinguished advantages and features. The Boland-Nicolaides trick is used in proving the inf-sup condition for the multidomain discrete problem. Optimal error estimates are derived for the coupled problem by analyzing the approximation errors and the consistency errors. Numerical examples are also provided to confirm the theoretical results.

A continuous finite element method to approximate Friedrichs' systems isproposed and analyzed. Stability is achieved by penalizing the jumpsacross meshinterfaces of the normal derivative of some components of the discrete solution. The convergence analysis leads to optimal convergence ratesin the graph norm and suboptimal of order ½ convergence rates inthe L 2-norm. A variant of the method specialized toFriedrichs' systems associated with elliptic PDE's in mixed form andreducing the number of nonzero entries in the stiffness matrix is alsoproposed and analyzed. Finally, numerical results are presented to illustrate thetheoretical analysis.