Expanded Mixed Finite Element Domain Decomposition Methods on Triangular Grids

In this work, we present a cell-centered time-splitting technique for solving evolutionary diffusion equations on triangular grids. To this end, we consider three variables (namely the pressure, the flux and a weighted gradient) and construct a so-called expanded mixed finite element method. This method introduces a suitable quadrature rule which permits to eliminate both fluxes and gradients, thus yielding a cell-centered semidiscrete scheme for the pressure with a local 10-point stencil. As for the time integration, we use a domain decomposition operator splitting based on a partition of unity function. Combining this splitting with a multiterm fractional step formula, we obtain a collection of uncoupled subdomain problems that can be efficiently solved in parallel. A priori error estimates for both the semidiscrete and fully discrete schemes are derived on smooth triangular meshes with six triangles per internal vertex.