The stabilized three-field domain decomposition method

In this work we consider the Three-Field finite element formulation for elliptic problems. In the
original setting, not all combinations of spaces will lead to stable formulations, and two independent
inf-sup conditions must be satisfied. One way to relax such constraint is to introduce stabilizations
terms, allowing more space choices. Our goal is to propose a stabilized scheme that allows different
combinations of polynomials for the unknowns involved. We explore here the possibility of employing

the three-field formulation as a direct method, i.e., no submeshes involved. We show coercivity
and convergence results in a suitable mesh dependent norm. Efficient implementation of the method
is still possible, as static condensation of the unknowns can be performed at the element level, in
parallel. We present numerical results displaying the performance of the method.