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：在数值模拟中，流体所占据的光滑域必须用通常与物理域不一致的计算域来近似。因此，为研究数值方法的收敛性与误差估计，需要考虑与域相关的离散误差，即所谓的变分犯罪。本文通过惩罚方法提出了一种优雅的替代方案，用以替代对变分犯罪进行直接但较为繁琐的分析。我们将物理域嵌入一个足够大的立方体域中，并研究相应域惩罚问题的有限体积方法的收敛性。我们证明，惩罚问题的数值解收敛于原始问题的广义解，即所谓的耗散弱解。若存在强解，则相同初始数据产生的耗散弱解与该强解一致。在此情形下，我们应用相对能量这一新工具，推导出数值解与强解之间的误差估计。文中还展示了大量数值实验，以验证理论结果。