Error analysis of a DG method employing ideal elements applied to a nonlinear convection–diffusion problem

Abstract

In this paper we use the discontinuous Galerkin finite element method for the space-semidiscretization of a nonlinear nonstationary convection–diffusion problem defined on a nonpolygonal two-dimensional domain. Using Zlámal's concept of the ideal curved elements, we define a finite element space . We prove the ‘ideal’ versions of the inverse and the multiplicative trace inequalities known for standard straight triangulations. Further, we define a projection on the finite element space and study its approximation properties. The obtained results allow us to derive an H¹-optimal error estimate for the discontinuous Galerkin method employing the ideal curved elements.