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A two-grid method for finite volume element approximations of second-order nonlinear hyperbolic equations. (English) Zbl 1190.65150
The authors consider a second order nonlinear hyperbolic equation. A semidiscrete finite volume element method, based on the two-grid method, is suggested and analyzed. The idea of the two grid method is to reduce the nonlinear and nonsymmetric problem on a fine grid into a linear and symmetric problem on a coarse grid. The basic mechanisms are two quasi uniform triangulations of $\Omega$, $T_H$ and $T_h$, with two different sizes $H$ and $h$ ($H>h$), and the corresponding finite element spaces $V_H$ and $V_h$ which satisfy $V_H\subset\,V_h$. An $H^1$ error estimate of order $h+H^3\log\vert\,H\vert$ is proved. A numerical test is presented to justify the efficiency of the method.

##### MSC:
 65M55 Multigrid methods; domain decomposition (IVP of PDE) 65M08 Finite volume methods (IVP of PDE) 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) 65M20 Method of lines (IVP of PDE) 35L70 Nonlinear second-order hyperbolic equations 65M15 Error bounds (IVP of PDE)
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