To solve flow problems associated with the Navier-Stokes equations, we construct a mixed finite volume/finite element method for the spatial approximation of the convective and diffusive parts of the flux, respectively. The finite volume component of the method is adapted from the authors' construction, for hyperbolic conservation laws and rectangular or unstructured triangular grids, of two-dimensional finite volume extensions of the Lax-Friedrichs and Nessyahu-Tadmor central difference schemes, in which the resolution of Riemann problems at cell interfaces is bypassed thanks to the use of the Lax-Friedrichs scheme on two specific staggered grids. MUSCL-type piecewise linear cell interpolants, slope limiters, and a two-step predictor-corrector time discretization lead to an oscillation-free quasi-second-order resolution. For the viscous terms, we use a centered finite element approximation inspired by Rostand-Stoufflet. To improve the quality of the resolution, we use a grid adaptation algorithm proposed by Castro Diaz and Hecht