On a semi-variational method for parabolic equations. I

summary:The paper aims at a further development of the finite element method, when applied to mixed problems for parabolic equations. Much work has been done on a special Galerkin-type procedure of order $\tau^2$, which is similar to the Crank-Nicholson finite-difference scheme. Here a sequence of approximations is presented, possessing an increasing accuracy in the time increment $\tau$. The first approximation coincides with the above-mentioned procedure. For the second approximation, the rate of convergence $\tau^4$ and the stability with respect to the initial condition is proved. The efficiency of the first and second approximations are compared on a numerical example