Abgrall R, Nordström J, Öffner P, Tokareva S
Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland.
Department of Mathematics, Computational Mathematics, Linköping University, 581 83 Linköping, Sweden.
J Sci Comput. 2020;85(2):43. doi: 10.1007/s10915-020-01349-z. Epub 2020 Nov 3.
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.
在双曲型问题领域，当考虑有限元方法时，间断伽辽金（DG）方法被广泛应用。顾名思义，DG框架允许在单元界面处存在间断，这对于许多研究人员来说，在处理双曲型平衡律问题时是一个有利的特性。相反，连续伽辽金方法似乎不适用于双曲型问题，并且仍然存在一种观念，即连续伽辽金方法极其不稳定。为了解决这个问题，通常会添加稳定项，并且在文献中可以找到各种不同的公式。然而，这种观念并不正确，一般来说稳定项是不必要的。在本文中，我们处理这个问题，但提出了一种不同的方法。我们按照有限差分领域常用的步骤，利用边界条件来稳定格式。这里的主要思想是弱施加边界条件，并构造特定的边界算子以确保稳定性。这种方法已经在间断伽辽金框架中使用过，但在这里我们将其应用于连续伽辽金格式。即使使用非结构化网格，也不需要内部耗散。此外，我们指出不需要精确积分，只要求积规则和微分算子中的范数相同就足够了，这样分部求和性质就成立，这意味着离散高斯定理是有效的。这与双曲型问题领域中认为纯伽辽金格式存在稳定性问题的观念相矛盾。在数值模拟中，我们验证了我们的理论分析。
