We establish stable finite element (FE) approximations of convection-diffusion initial boundary value problems using the automatic variationally stable finite element (AVS-FE) method. The transient convection-diffusion problem leads to issues in classical FE methods as the differential operator can be considered singular perturbation in both space and time. The unconditional stability of the AVS-FE method, regardless of the underlying differential operator, allows us significant flexibility in the construction of FE approximations. We take two distinct approaches to the FE discretization of the convection-diffusion problem: i) considering a space-time approach in which the temporal discretization is established using finite elements, and ii) a method of lines approach in which we employ the AVS-FE method in space whereas the temporal domain is discretized using the generalized-alpha method. In the generalized-alpha method, we discretize the temporal domain into finite sized time-steps and adopt the generalized-alpha method as time integrator. Then, we derive a corresponding norm for the obtained operator to guarantee the temporal stability of the method. We present numerical verifications for both approaches, including numerical asymptotic convergence studies highlighting optimal convergence properties. Furthermore, in the spirit of the discontinuous Petrov-Galerkin method by Demkowicz and Gopalakrishnan, the AVS-FE method also leads to readily available a posteriori error estimates through a Riesz representer of the residual of the AVS-FE approximations. Hence, the norm of the resulting local restrictions of these estimates serve as error indicators in both space and time for which we present multiple numerical verifications adaptive strategies.

翻译：我们采用自适应变分稳定有限元法建立了对流-扩散初边值问题的稳定有限元逼近。由于微分算子在空间和时间上均可视为奇异摄动，瞬态对流-扩散问题在经典有限元方法中会引发挑战。AVS-FE方法无条件稳定的特性（不受底层微分算子影响）使我们在构建有限元逼近时具有显著的灵活性。本文采用两种不同方法对该问题实施有限元离散：(i) 时空方法，通过有限元建立时间离散；(ii) 线方法，在空间上应用AVS-FE法，而时间域采用广义α法离散。在广义α法中，我们将时间域离散为有限尺寸的时间步长，并采用广义α法作为时间积分器，进而推导相应算子范数以保障方法的时间稳定性。我们通过数值验证两种方法，包括展示最优收敛特性的数值渐近收敛性研究。此外，遵循Demkowicz与Gopalakrishnan提出的不连续Petrov-Galerkin法理念，AVS-FE法还可通过残差的Riesz表示算子直接获得后验误差估计。这些估计的局部限制范数可作为空间与时间上的误差指示子，我们据此展示了多项自适应策略的数值验证结果。