In numerical simulations a smooth domain occupied by a fluid has to be approximated by a computational domain that typically does not coincide with a physical domain. Consequently, in order to study convergence and error estimates of a numerical method domain-related discretization errors, the so-called variational crimes, need to be taken into account. In this paper we present an elegant alternative to a direct, but rather technical, analysis of variational crimes by means of the penalty approach. We embed the physical domain into a large enough cubed domain and study the convergence of a finite volume method for the corresponding domain-penalized problem. We show that numerical solutions of the penalized problem converge to a generalized, the so-called dissipative weak, solution of the original problem. If a strong solution exists, the dissipative weak solution emanating from the same initial data coincides with the strong solution. In this case, we apply a novel tool of the relative energy and derive the error estimates between the numerical solution and the strong solution. Extensive numerical experiments that confirm theoretical results are presented.
翻译:在数字模拟中,流体所占据的光滑域必须用通常与物理域不相吻合的计算域加以近似。 因此,为了研究数字方法域相关离散误的趋同和误差估计值,需要考虑到所谓的变异犯罪。 在本文中,我们提出了一个优雅的替代方法,而不是直接分析,而是技术性的,通过惩罚方法对变异犯罪进行分析。 我们将物理域嵌入一个足够大的立方体域域,并研究相应域化问题的有限体积方法的趋同。 我们显示,受罚问题的数字解决办法将集中到一个普遍化的、所谓的消散弱、解决原始问题的办法。 如果存在强有力的解决办法,来自同一初始数据的消散弱解决办法与强有力的解决办法相吻合。 在这种情况下,我们应用了相对能量的新工具,并在数字解决办法和强型解决办法之间得出误差估计值。我们提出了广泛的数字实验,以证实理论结果。