In this paper, we present convergence analysis of high-order finite element based methods, in particular, we focus on a discontinuous Galerkin scheme using summation-by-parts operators. To this end, it is crucial that structure preserving properties, such as positivity preservation and entropy inequality hold. We demonstrate how to ensure them and prove the convergence of our multidimensional high-order DG scheme via dissipative weak solutions. In numerical simulations, we verify our theoretical results.

翻译：在本文中,我们提出对基于高阶有限要素的方法的趋同分析,特别是,我们注重采用按部逐级的操作员对Galerkin计划进行不连续性的Galerkin计划,为此,必须建立保护特性的结构,例如保护活性、保护活性、不不平等等结构。我们展示如何确保这些特性,并证明我们多层面的高阶DG计划通过分散式的薄弱解决方案实现趋同。在数字模拟中,我们核查我们的理论结果。