Backward Euler method for 2D Sobolev equation with Burgers’ type non-linearity

Backward Euler for two dimensional Sobolev equation is discussed in this article. We begin by obtaining some basic a priori estimates for the semi-discrete scheme and for the backward Euler approximation. It is proven that these estimations for the discrete scheme are valid uniformly in time using the discrete Gronwall’s Lemma. In addition, the presence of a discrete global attractor is established. Furthermore, optimal a priori error bounds are determined, which are time dependent exponentially. Under the uniqueness condition, these error estimates are demonstrated to be uniform in time. Finally, we establish several numerical examples that validate our theoretical approach.