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

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.