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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 errors are of second-order.

MSC:
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
35L75Nonlinear hyperbolic PDE of higher (>2) order
65M15Error bounds (IVP of PDE)
35Q51Soliton-like equations
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76X05Ionized gas flow in electromagnetic fields; plasmic flow
76M10Finite element methods (fluid mechanics)