Second order backward difference scheme combined with finite element method for a 2D Sobolev equation with Burgers' type non-linearity

In this article, a second-order backward difference scheme combined with a conforming finite element method (FEM) is applied to a two-dimensional Sobolev equation with Burgers' type non-linearity with a nonhomogeneous forcing function in L-infinity(L-2). Some new a prioriestimates are derived for the semidiscrete solution, which helps to prove the convergence result. Then, based on the Stolz-Cesaro theorem, a prioribounds for the second-order backward difference scheme are derived, which are valid uniformly in time as t(N) -> infinity and uniformly in the dispersion coefficient mu as mu -> 0. For the discrete problem at each time step, it is shown using one variant of Brouwer's fixed point theorem that, there exists a discrete solution and uniqueness follows in a standard way. Moreover, existence of a discrete global attractor is established. Optimal error estimates in l(infinity)(l(2)) and l(infinity)(H-0(1))-norms are obtained, whose bounds may depend exponentially on time. Subsequently, error bounds are established, which are valid uniformly in time under uniqueness conditions. Additionally, it is shown that as mu -> 0, the completely discrete solution of the 2D Sobolev equation converges to the corresponding discrete solution of the Burgers' equation. Further, results on extrapolated Backward difference scheme are briefly discussed. Finally, some computational experiments are conducted, whose results confirm our theoretical findings.