Finite element Galerkin method for 2D Sobolev equations with Burgers’ type nonlinearity

摘要

In this article, the global existence of a unique strong solution to the 2D Sobolev equation with Burgers’ type nonlinearity is established using weak or weak* compactness type arguments. When the forcing function (f ≠ 0) is in L∞(L2), new a priori bounds are derived, which are valid uniformly in time as t↦∞ and with respect to the dispersion coefficient μ as μ↦0. It is further shown that the solution of the Sobolev equation converges to the solution of the 2D-Burgers’ equation with order O(μ). A finite element method is applied to approximate the solution in the spatial direction and the existence of a global attractor is derived for the semidiscrete scheme. Further, using a priori bounds and an integral operator, optimal error estimates are derived in L∞(L2)-norm, which hold uniformly with respect to μ as μ → 0. Since the constants in the error estimates have exponential growth in time, therefore, under a certain uniqueness condition, the error bounds are derived which are uniformly in time. More importantly, all the above results remain valid as μ tends to zero. Finally, this paper concludes with some numerical examples.