Three cases for the nonlinear Sobolev equation c(x,u)(Ou/Ot)-V.(a(x,u)Vu+ b(x, u, Vu)V(Ou/Ot))=f(x, t, u, Vu) are studied. In case I, the coefficients a and b have uniform positive lower bounds in a neighborhood of the solution; in case II, b b(x, u) is allowed to take zero values and possibly cause the Sobolev equation to degenerate to a parabolic equation; in case III we only require a bound of the form la(x, u)l<K with a positive lower bound on b b(x, u, Vu). A Crank-Nicolson-Galerkin approximation with extrapolated coefficients is presented for all cases along with a conjugate gradient iterative procedure which can be used efficiently to solve the different linear systems of algebraic equations arising at each step from the Galerkin method. A priori error estimates are derived for each approximation. Optimal order Hi-error estimates are obtained in each case.