Unconditional superconvergence analysis of conforming finite element for nonlinear parabolic equation

Galerkin finite element approximation to nonlinear parabolic equation is studied with a linearized backward Euler scheme. The error between the exact solution and the numerical solution is split into two parts which are called the temporal error and the spatial error through building a time-discrete system. On one hand, the temporal error derived skillfully leads to the regularity of the time-discrete system solution. On the other hand, the τ-independent spatial error and the boundedness of the numerical solution in L∞-norm is deduced with the above achievements. At last, the superclose result of order O(h2+τ)O(h2+τ) in H1-norm is obtained without any restriction of τ in a routine way. Here, h is the subdivision parameter, and τ, the time step.