Curved Elements in the Finite Element Method. I

Summary:The paper deals with an initial problem of a parabolic variational inequality whichcontains a nonlinear elliptic form $a(v,w)$ having a potential $J(v)$, which is twice $G$-differentiable at arbitrary $v\in H^1(\Omega)$. This property of $a(v,w)$ makes it possible to prove convergence of an approximate solution defined by a linearized scheme which is fully discretized - in space by the finite elements method and in time by a one-step finite-difference method. Strong convergence of the approximate solution is proved without any regularity assumptions on the exact solution. An error bound is also derived under the assumption that the exact solution is sufficiently smooth