We study the convergence and error estimates of a finite volume method for the compressible Navier-Stokes-Fourier system with Dirichlet boundary conditions. Physical fluid domain is typically smooth and needs to be approximated by a polygonal computational domain. This leads to domain-related discretization errors, the so-called variational crimes. To treat them efficiently we embed the fluid domain into a large enough cubed domain, and propose a finite volume scheme for the corresponding domain-penalized problem. Under the assumption that the numerical density and temperature are uniformly bounded, we derive the ballistic energy inequality, yielding a priori estimates and the consistency of the penalization finite volume approximations. Further, we show that the numerical solutions converge weakly to a generalized, the so-called dissipative measure-valued, solution of the corresponding Dirichlet problem. If a strong solution exists, we prove that our numerical approximations converge strongly with the rate 1/4. Additionally, assuming uniform boundedness of the approximate velocities, we obtain global existence of the strong solution. In this case we prove that the numerical solutions converge strongly to the strong solution with the optimal rate 1/2.

翻译：我们研究了带有Dirichlet边界条件的可压缩Navier-Stokes-Fourier系统的有限体积方法的收敛性与误差估计。物理流体域通常是光滑的，需要用多边形计算域进行近似，这导致了与域相关的离散误差，即所谓的变分犯罪。为了高效处理这些误差，我们将流体域嵌入一个足够大的长方体域中，并提出相应的域惩罚问题的有限体积格式。在数值密度和温度一致有界的假设下，我们推导出弹性能量不等式，从而得到先验估计和惩罚有限体积近似的一致性。进一步地，我们证明数值解弱收敛到对应Dirichlet问题的一个广义解，即所谓的耗散测度值解。若存在强解，我们证明数值近似以1/4的速率强收敛。此外，假设近似速度一致有界，我们得到了强解的全局存在性。在这种情况下，我们证明数值解以最优速率1/2强收敛到强解。