Mixed finite element methods and tree-cotree implicit condensation

Mixed methods are widely used for the finite element analysis of differential problems. From a computational point of view, however, the increase in the number of variables from that of the original problem and the properties of the resulting matrices often make a direct implementation of these formulations inadvisable. In this paper, the mixed formulation of Dirichlet’s problem for the Laplace operator, discretised using lowest degree Raviart–Thomas elements will be considered as an example. A tree-cotree decomposition of the graph associated with the mesh allows the implemen- tation of an equivalent method, which significantly reduces the number of degrees of freedom and gives rise to a symmetric positive definite linear system. We will refer to this novel method as tcic (Tree-Cotree Implicit Condensation). Numerical results will be presented, and the possibility of extending the proposed method to other cases will be discussed.