We present a space-time multiscale method for a parabolic model problem with an underlying coefficient that may be highly oscillatory with respect to both the spatial and the temporal variables. The method is based on the framework of the Variational Multiscale Method in the context of a space-time formulation and computes a coarse-scale representation of the differential operator that is enriched by auxiliary space-time corrector functions. Once computed, the coarse-scale representation allows us to efficiently obtain well-approximating discrete solutions for multiple right-hand sides. We prove first-order convergence independently of the oscillation scales in the coefficient and illustrate how the space-time correctors decay exponentially in both space and time, making it possible to localize the corresponding computations. This localization allows us to define a practical and computationally efficient method in terms of complexity and memory, for which we provide a posteriori error estimates and present numerical examples.

翻译：我们为抛物线模型问题提出了一个时时多尺度方法,其基本系数在空间变量和时间变量方面可能都是高度随机的。该方法基于空间时间配制背景下的变异多尺度方法框架,并计算出由辅助空间-时间校正功能充实的不同操作员的粗尺度代表。一旦计算出来,粗尺度的表示使我们能够有效地为多个右侧获得非常相近的离散解决办法。我们证明,与该系数的振荡尺度无关,第一阶的趋同性一致,并说明了空间-时间校正器如何在空间和时间两方面加速衰减,从而有可能将相应的计算方法本地化。这种局部化使我们能够界定一个在复杂性和记忆方面实用和计算高效的方法,为此,我们提供了事后误差估计,并提供了数字实例。