We address the issue of the suboptimality in the p-version discontinuous Galerkin (dG) methods for first order hyperbolic problems. The convergence rate is derived for the upwind dG scheme on tensor product meshes in any dimension. The standard proof in seminal work [14] leads to suboptimal convergence in terms of the polynomial degree by 3/2 order for general convection fields, with the exception of piecewise multi-linear convection fields, which rather yield optimal convergence. Such suboptimality is not observed numerically. Thus, it might be caused by a limitation of the analysis, which we partially overcome: for a special class of convection fields, we shall show that the dG method has a p-convergence rate suboptimal by 1/2 order only.

翻译：我们处理Pversion不连续的 Galerkin (dG) 方法中用于一阶双曲问题的亚最佳度问题。 调合率是针对任一维度的高压制成的上风 dG 方法的。 初级工作[14] 的标准证明导致一般对流字段多角度3/2顺序的亚最佳趋同, 但小巧多线性对流字段除外, 后者更能产生最佳的趋同。 这种次优性在数字上没有被观察到。 因此, 这可能是由于分析的局限性造成的, 我们部分克服了这种局限性: 对于特殊的对流字段, 我们只能证明 dG 方法有 p- 趋同率 1/2 的亚优度 。