A Cartesian, cell-based approach for adaptively-refined solutions of the Euler and Navier-Stokes equations

Discrete Sobolev inequalities andLperror estimates for finite volume solutions of convection diffusion equations

“…For more general meshes (see Def. 1), we give an extension of the previous scheme which is successfully used in practice [5,21] and has been proved to converge on quadrangular meshes [9], on rectangular meshes with local refinement [10], and on admissible meshes (in which case it is identical to the previous scheme). Section 4 is concerned with the proof of some discrete inequalities of Sobolev for functions defined on general meshes which yield the final L p error estimates for the schemes.…”

Convergence rate of a finite volume scheme for the linear convection-diffusion equation on locally refined meshes

“…Indeed, under the assumption above, such a constant exists in any of the reference cases: then the scheme defined by (3) to (8), and the weights of Figure 5 is coercive.…”

“…where the numerical fluxes are defined by (3) to (8). This operator is not consistent in general (in the sense of finite differences, see [17,22]).…”

“…where φ D e (ū h ) is given by (5), (6), (7), or (8), assuming that g = u |Γ D and k = ∂ n u |Γ N . We recall that u h is the L 2 projection of u:ū Proof.…”

“…where L D h u h is calculated according to (5)(6)(7)(8), with the boundary conditions k = 0 and g = 0 (see Sect. 2).…”