A new error estimates of finite element method for (2+1)-dimensional nonlinear advection-diffusion model

Abstract

This paper designs, analyzes, and implements a finite element analysis for the simulation of the nonlinear advection-diffusion model. The key feature of the proposed work is to provide novel a priori error estimates in Bôchner norms L2(0,T;H01(Ω)) and L∞(0,T;H01(Ω)) simultaneously. To achieve the desired error estimates for both semidiscrete and fully discrete schemes, our idea is to introduce the Ritz projection operator that allows us to enforce the known estimates for P1 and P2 finite elements in the formulation of the desired results. Further, this paper establishes the stability, existence, and uniqueness in the precise normed spaces. Finally, we prove the fully discrete error estimates using the Backward Euler and Crank-Nicolson's schemes which are corroborated by numerical experiments to validate the developed theory on various domains for both schemes. To the best of author's knowledge, this is the first finite element simulation that provides the estimates in the L2 Bôchner norm for any such types of time-dependent problems.