We carry out a stability and convergence analysis for the fully discrete scheme obtained by combining a finite or virtual element spatial discretization with the upwind-discontinuous Galerkin time-stepping applied to the time-dependent advection-diffusion equation. A space-time streamline-upwind Petrov-Galerkin term is used to stabilize the method. More precisely, we show that the method is inf-sup stable with constant independent of the diffusion coefficient, which ensures the robustness of the method in the convection- and diffusion-dominated regimes. Moreover, we prove optimal convergence rates in both regimes for the error in the energy norm. An important feature of the presented analysis is the control in the full $L^2(0,T;L^2(\Omega))$ norm without the need of introducing an artificial reaction term in the model. We finally present some numerical experiments in $(3 + 1)$-dimensions that validate our theoretical results.

翻译：本文针对含时对流扩散方程，结合有限元或虚拟元空间离散与迎风间断伽辽金时间步进法得到的全离散格式，进行了稳定性与收敛性分析。该方法通过引入时空流线迎风彼得罗夫-伽辽金项实现稳定化。具体而言，我们证明了该方法具有与扩散系数无关的inf-sup稳定常数，从而确保方法在对流主导和扩散主导区域均保持鲁棒性。此外，我们证明了在能量范数意义下，两种区域均能达到最优收敛阶。本分析的一个重要特点是在不引入人工反应项的前提下，实现了对全$L^2(0,T;L^2(\Omega))$范数的控制。最后，我们通过$(3 + 1)$维数值实验验证了理论结果。