Cell centered Galerkin methods for diffusive problems (Q2842488)

The author introduces a new class of methods for diffusive problems on general meshes with only one unknown per element in order to devise a suitable framework for a multi-physics platform based on lowest-order methods. The main idea, inspired by discontinuous Galerkin methods, is to construct an incomplete piecewise affine polynomial space with optimal approximation properties starting from values at cell centers. The author considers in particular cell centered Galerkin methods with cell unknowns only, i.e., the unknown in each mesh element is interpreted as the value of the discrete function at a given point (the cell center). A discrete space \(V_{h}^{ccg}\) is used as a test/trial space in a suitable nonconforming finite element setting. To infer convergence rates, the author studies the approximation properties of the space \(V_{h}^{ccg}\) with respect to the energy norm naturally associated to the discrete problem. Two applications are considered: a homogeneous anisotropic scalar diffusion problem (which offers a simplified context to outline the main ideas of the method) and the steady incompressible Navier-Stokes equations.