Solving Convection-Diffusion Problems on Curved Domains by Extensions from Subdomains

Abstract

We present a technique for numerically solving convection-diffusion problems in domains $\varOmega$ with curved boundary. The technique consists in approximating the domain $\varOmega$ by polyhedral subdomains $\mathsf{{D}}_h$ where a finite element method is used to solve for the approximate solution. The approximation is then suitably extended to the remaining part of the domain $\varOmega$. This approach allows for the use of only polyhedral elements; there is no need of fitting the boundary in order to obtain an accurate approximation of the solution. To achieve this, the boundary condition on the border of $\varOmega$ is transferred to the border of $\mathsf{D }_h$ by using simple line integrals. We apply this technique to the hybridizable discontinuous Galerkin method and provide extensive numerical experiments showing that, whenever the distance of $\mathsf{{D}}_h$ to $\partial \varOmega$ is of order of the meshsize $h$, the convergence properties of the resulting method are the same as those for the case in which $\varOmega =\mathsf{{D}}_h$. We also show numerical evidence indicating that the ratio of the $L^2(\varOmega )$ norm of the error in the scalar variable computed with $d>0$ to that of that computed with $d=0$ remains constant (and fairly close to one), whenever the distance $d$ is proportional to $\min \{h,Pe^{-1}\}/(k+1)^2$, where $Pe$ is the so-called Péclet number.