Lax theorem and finite volume schemes

This work addresses a theory of convergence for finite volume methods applied to linear equations. A non-consistent model problem posed in an abstract Banach space is proved to be convergent. Then various examples show that the functional framework is non-empty. Convergence with a rate $h^{\frac12}$ of all TVD schemes for linear advection in 1D is an application of the general result. Using duality techniques and assuming enough regularity of the solution, convergence of the upwind finite volume scheme for linear advection on a 2D triangular mesh is proved in $L^\alpha$, $2\leq \alpha\leq +\infty$: provided the solution is in $W^{1,\infty}$, it proves a rate of convergence $h^{\frac14 -\varepsilon}$ in $L^\infty$.