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Lax theorem and finite volume schemes. (English) Zbl 1053.65073
The author presents an abstract framework for the numerical approximation of linear equations in general Banach spaces by means of finite volume methods. The present work aims at explaining the phenomena that for finite volume methods for linear equations which are convergent, consistency is not necessary. He proves that a non-consistent model problem posed in an abstract Banach space is convergent. The convergence of the upward finite volume schemes on a 2D triangulation mesh is proved.

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L45 Initial value problems for first-order hyperbolic systems
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