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$H^1$-Galerkin mixed finite element method for the regularized long wave equation. (English) Zbl 1098.65096
The regularized long wave equation in one space dimension, a scalar valued third-order hyperbolic partial differential equation, is discretized with a mixed finite element method. For time discretization a first-order backward Euler scheme is used. The equation describes nonlinear dispersive waves and has solitary wave solutions. It is for example encountered in modelling shallow water waves or ion acoustic plasma waves. The paper describes an $H^1$-Galerkin mixed finite element method for the above equation. The numerical analysis yields existence and uniqueness of the semi-discrete and the fully discrete solution. Optimal error estimates are also established. It is shown that no Ladyshenskaya-Babuška-Brezzi consistency condition, as necessary for classical mixed finite element methods, is required to approximate simultaneously the solution and the vector valued flux. Numerical results illustrate for a test problem for which an analytical solution is given the properties of the method. It is shown that the $H^1$ error is of order one while $L^2$ and $L^\infty$ errors are of second-order.

##### MSC:
 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) 35L75 Nonlinear hyperbolic PDE of higher $(>2)$ order 65M15 Error bounds (IVP of PDE) 35Q51 Soliton-like equations 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76X05 Ionized gas flow in electromagnetic fields; plasmic flow 76M10 Finite element methods (fluid mechanics)
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