In the hyperbolic community, discontinuous Galerkin (DG) approaches are mainly applied when finite element methods are considered. As the name suggested, the DG framework allows a discontinuity at the element interfaces, which seems for many researchers a favorable property in case of hyperbolic balance laws. On the contrary, continuous Galerkin methods appear to be unsuitable for hyperbolic problems and there exists still the perception that continuous Galerkin methods are notoriously unstable. To remedy this issue, stabilization terms are usually added and various formulations can be found in the literature. However, this perception is not true and the stabilization terms are unnecessary, in general. In this paper, we deal with this problem, but present a different approach. We use the boundary conditions to stabilize the scheme following a procedure that are frequently used in the finite difference community. Here, the main idea is to impose the boundary conditions weakly and specific boundary operators are constructed such that they guarantee stability. This approach has already been used in the discontinuous Galerkin framework, but here we apply it with a continuous Galerkin scheme. No internal dissipation is needed even if unstructured grids are used. Further, we point out that we do not need exact integration, it suffices if the quadrature rule and the norm in the differential operator are the same, such that the summation-by-parts property is fulfilled meaning that a discrete Gauss Theorem is valid. This contradicts the perception in the hyperbolic community that stability issues for pure Galerkin scheme exist. In numerical simulations, we verify our theoretical analysis.

在双曲线社区中，当考虑有限元方法时，主要应用不连续Galerkin（DG）方法。顾名思义，DG框架允许元素界面处的不连续性，对于许多研究人员来说，在双曲线平衡定律的情况下似乎是有利的。相反，连续的Galerkin方法似乎不适合于双曲线问题，并且仍然存在这样的看法，即众所周知的连续Galerkin方法是不稳定的。为了解决这个问题，通常添加稳定化术语，并且可以在文献中找到各种公式。但是，这种感觉通常是不正确的，稳定化术语是不必要的。在本文中，我们处理了这个问题，但是提出了一种不同的方法。我们使用边界条件按照有限差分社区中经常使用的程序来稳定方案。在这里，主要思想是弱加边界条件，并构造特定的边界算子以确保稳定性。这种方法已经在不连续的Galerkin框架中使用，但是在这里我们将其与连续的Galerkin方案一起使用。即使使用非结构化网格，也无需内部耗散。此外，我们指出，我们不需要精确积分，只要正交规则和微分算子中的范数相同就足够了，从而满足了部分求和属性，这意味着离散高斯定理是有效的。这与双曲线社区的看法相反，即纯粹的Galerkin方案存在​​稳定性问题。