Convergence of H 1-Galerkin mixed finite element method for parabolic problems with reduced regularity on initial data

Abstract

We study the convergence of H1-Galerkin mixed finite element method for parabolic problems in one space dimension. Both semi-discrete and fully discrete schemes are analyzed assuming less regularity on initial data. More precisely, for the spatially discrete scheme, error estimates of order \(\mathcal{O}\)(h2t−1/2) for positive time are established assuming the initial function p0 ∈ H2(Ω) ∩ H01 (Ω). Further, we use energy technique together with parabolic duality argument to derive error estimates of order \(\mathcal{O}\)(h2t−1) when p0 is only in H01 (Ω). A discrete-in-time backward Euler method is analyzed and almost optimal order error bounds are established.