Finite element Galerkin method for the “good” Boussinesq equation

Abstract

Recently, it has been shown by Manoranjan et al. [l] that this class of equations possesses an interesting solitary wave interaction mechanism. Moreover using semi group theoretic approch they have established global existence of a generalized solution, provided the initial data are in a potential well. Further it is shown that finite time blowup may occur for weak solution in R, if the initial data are not in the potential well and the total energy at t = 0 exceeds the depth of the well. In the first part of this paper we shall discuss existence, uniqueness and regularity results for the problem (1 .I)-( 1.3) using the Faedo-Galerkin method. In the literature, very few results are available pertaining to the numerical study of (1 .l)-( 1.4). In [2], Defrutos et al. have analysed pseudo spectral Fourier method for the equation (I. I)- (I .2) with periodic boundary conditions. They have also discussed the nonlinear stability and convergence results in the analytic framework introduced earlier by Lopez, Marcos and Sanz- Serna [3]. For related results on finite difference schemes, one may refer to Ortega and Sanz- Serna [4]. Recently, Manoranjan et al. [5] have applied Petrov-Galerkin finite element method for Cauchy problem without rigorous error analysis and have presented some numerical results. In the later part of the present paper, we have derived optima1 error estimates for both semidiscrete