Superconvergence and asymptotic expansion for semidiscrete bilinear finite volume element approximation of the parabolic problem

We first derive the asymptotic expansion of the bilinear finite volume element for the linear parabolic problem by employing the energy-embedded method on uniform grids, and then obtain a high accuracy combination pointwise formula of the derivatives for the finite volume element approximation based on the above asymptotic expansion. Furthermore, we prove that the approximate derivatives have the convergence rate of order two. Numerical experiments confirm the theoretical results.