Numerical solution of pseudoparabolic equations (Q5970607)

The paper deals with equations of the form \[ (-u/t)+L(x)(-u/t)+M(x)u= f(t,x)\tag{1} \] considered in the cylindrical domain \([0,T]\times \overline\Omega\) with bounded \(\Omega\subset \mathbb{R}^n\). Here \(L\), \(M\) are linear differential operators of second order, \(L\) is uniformly elliptic in \(\overline\Omega\), \(M\) is positive and the right member \(f\) may be a generalized function. Equation (1) is provided with an initial condition and a boundary condition of Neumann type. Using the Galerkin method two sequences of the approximate solutions \(u_k(t,x)\) are constructed and their convergence to the exact solution (belonging to a Sobolev space) is proved, using the so-called ``negative spaces'' [see \textit{Yu. M. Berezanskij}, Eigenfunction expansion of selfadjoint operators (Russian), Naukova Dumka, Kiev (1965; Zbl 0142.37203)].