The mixed finite element approximation of the Poisson problem leads to an indefinite linear system with a saddle point structure. Although a large selection of iterative methods tailored for solving such problems can be found from the literature, their success depends completely on the availability of a good preconditioner. The existing preconditioners for such problems are constructed either by approximating the flux or the pressure solution operators. Preconditioners in the latter group take advantage of the equivalence between a nonconforming finite element method and the pressure solution operator. These equivalences lead to the design and analysis of preconditioners for the nonconforming problem.
We will explore the use of preconditioners designed for piecewise linear finite element discretizations of the Laplace operator as preconditioners for the mixed problem. Combined with a suitable iterative method, the number of iterations required to solve the preconditioned system will have the same mesh size dependency as for the preconditioner applied to the original Poisson problem. The main benefit of the presented approach is in the possibility to reuse existing Laplace preconditioners, i.e. in the ease of implementation.