A curved boundary treatment for discontinuous Galerkin schemes solving time dependent problems

For problems defined in a two-dimensional domain Ω with boundary conditions specified on a curve Γ, we consider discontinuous Galerkin (DG) schemes with high order polynomial basis functions on a geometry fitting triangular mesh. It is well known that directly imposing the given boundary conditions on a piecewise segment approximation boundary Γh will render any finite element method to be at most second order accurate. Unless the boundary conditions can be accurately transferred from Γ to Γh, in general curvilinear element method should be used to obtain high order accuracy. We discuss a simple boundary treatment which can be implemented as a modified DG scheme defined on triangles adjacent to Γh. Even though integration along the curve is still necessary, integrals over any curved element are avoided. If the domain Ω is convex, or if Ω is nonconvex and the true solutions can be smoothly extended to the exterior of Ω, the modified DG scheme is high order accurate....

• Type: Journal
• Year: 2016
• Where: JCPHY
• Authors: Xiangxiong Zhang
• Keywords: JCPHY 2016