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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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