A weak Galerkin finite element method for nonlinear convection-diffusion equation

Abstract

In this paper, a weak Galerkin (WG) finite element method for one-dimensional nonlinear convection-diffusion equation with Dirichlet boundary condition is developed. Based on a special variational form featuring two built-in parameters, the semi-discrete and fully discrete WG finite element schemes are proposed. The backward Euler method is utilized for time discretization. The WG finite element method adopts locally piecewise polynomials of degree k for the approximation of the primal variable in the interior of elements, and piecewise polynomials of degree k+1 for the weak derivatives. Theoretically, the optimal error estimates in both discrete H1 and standard L2 norms are derived. Numerical experiments are performed to demonstrate the effectiveness of the WG finite element approach and validate the theoretical findings.