The problem of construction of the accurate methods, on the basis of finite differences, is closely connected with the improvement of the accuracy of numerical solutions in the neighborhood of a discontinuity point. The localization of such points is essentially simplified, if the difference method used enables us to reduce the width of deformation area to a single spatial increment of the discrete grid. In such case one can apply, for instance, procedures determining the maximum gradient of any sought-for function, which becomes in a certain region strong irregular or even discontinuous. This means that the maximum value of the first derivative of such a function must be determined in one spatial increment of the grid. The finite element method in Hermitian spaces seems to be one of the best suited for such purposes. Some finite element approximations and difference schemes for the linear and non-linear equation will be analyzed and some numerical solutions will be presented.