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We prove the local decay of the energy of the solution to a mixed initial boundary value problem for the linearized shallow‐water equations with constant coefficients, where the domain is a half‐plane, a certain dissipative boundary condition is prescribed and the initial data have compact support contained in the open half‐plane.

Abstract. In this paper we consider certain semidiscrete and fully discrete Galerkin approximations to the solution of an initial-boundary value problem for a second-order hyperbolic equation with a dissipative term. Estimates are obtained in the energy and negative norms associated with the problem, yielding in particular //'-and L2-error estimates. The approximation to the initial data is taken, in this case, as the projection with respect to the energy inner product, onto the approximating space. We also obtain estimates for higher-order time derivatives.