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$$L^\infty$$- and $$L^2$$-error estimates for a finite volume approximation of linear advection. (English) Zbl 1171.35008
The author studies the convergence of the upwind finite volume scheme applied to the linear advection equation
$\begin{cases} u_t+ \text{div} \left( Vf\left( u\right) \right) =0\quad &\text{on }{\mathbb{R}}_x^d\times {\mathbb{R}}_{+t}, \\ u\left( x,0\right) =u_0& \text{on }{\mathbb{R}}^d, \end{cases}$
where the vector $$V$$ is a Lipschitz function. One demonstrates a $$h^{1/2-\varepsilon }$$- error estimate in the $$L^\infty \left( {\mathbb{R}} ^d\times \left[ 0,T\right] \right)$$ - norm for Lipschitz initial data. One also proves a $$h^{1/2}$$ - error estimate in the $$L^\infty \left( 0,T;L^2\left( {\mathbb{R}}^d\right) \right)$$ - norm for initial data in $$H^1\left( {\mathbb{R}}^d\right) .$$

##### MSC:
 35A35 Theoretical approximation in context of PDEs 35L65 Hyperbolic conservation laws 65M15 Error bounds for initial value and initial-boundary value problems involving PDEs 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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