The characteristic finite volume element method for the nonlinear convection-dominated diffusion problem

In modern numerical simulation of prospecting and exploiting oil–gas resources and environmental science, it is important to consider a numerical method for nonlinear convection-dominated diffusion problems. Based on actual conditions, such as the three-dimensional characteristics of large-scale science-engineering computation, we present a kind of characteristic finite volume element method. Some techniques, such as calculus of variations, commutating operators, the theory of prior estimates and techniques, are adopted. Suboptimal order error estimate in L2 norm and optimal order error estimate in H1 norm are derived to determine the errors for the approximate solution. Numerical results are presented to verify the performance of the scheme.