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A discontinuous Galerkin method for the Rosenau equation. (English) Zbl 1145.65071
This paper is concerned with a posteriori error estimates for the Rosenau equation $$u_t+u_{xxxxt}=f(u)_x$$ in $\Omega\times (0,T]$ subject to the boundary conditions $u(x,t)=u_x(x,t)=0$ on $\partial\Omega\times (0,T]$ and $u(x,0)=u_0(x)$, $x\in\overline\Omega$. Here $\Omega=(0,1)$, $f(u)=-u-u^2$ and $T>0$. A posteriori estimates are obtained by using the discontinuous Galerkin method. The stability of the approximated solution is also discussed. Numerical results on a posteriori error and wave propagation are given, which are obtained by using various spatial and temporal meshes controlled automatically by a posteriori error.

##### MSC:
 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) 65M12 Stability and convergence of numerical methods (IVP of PDE) 65M15 Error bounds (IVP of PDE) 35K55 Nonlinear parabolic equations
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