A priori error estimates of expanded mixed FEM for Kirchhoff type parabolic equation

Abstract

For a nonlinear nonlocal parabolic problem containing the elastic energy coefficients, an expanded mixed finite element method using lowest order RT spaces is discussed in this paper. Firstly, some new regularity results are derived avoiding compatibility conditions on the data, which reflect behavior of exact solution as t → 0. Then, a semidiscrete method is derived on applying expanded mixed scheme in spatial direction keeping time variable continuous. A priori estimates for the discrete solutions are discussed under appropriate regularity assumptions and a priori error estimates in L^∞(L²(Ω)) norm for the solution, the gradient and its flux are established for both the semidicsrete and fully discrete system, when the initial data is in $H^{2}({\Omega }) \cap {H^{1}_{0}}({\Omega })$. Based on the backward Euler method, a completely discrete scheme is derived and existence of a unique fully discrete numerical solution is proved by using a variant of Brouwer’s fixed point theorem. Then, the corresponding error analysis is established. Further, numerical experiments are conducted for confirming our theoretical results.