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

摘要

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 Zlamal's concept of the ideal curved elements, we define a finite element space S_(hp). 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 S_(hp) and study its approximation properties. The obtained results allow us to derive an H~1 -optimal error estimate for the discontinuous Galerkin method employing the ideal curved elements. 【Keywords】 nonlinear convection-diffusion equation; discontinuous Galerkin finite element method; curved boundary; ideal triangulation; error estimates;