AI 中文总结
研究抛物问题的时空伽辽金 - 彼得罗夫格式，分析其与经典隐式时间步长格式关系及条件稳定性，通过重新审视连续伽辽金方法进行误差分析，并推导伴随时空格式，理论与实验结合得出结果。
AI 中文摘要
我们研究抛物型演化问题的时空伽辽金 - 彼得罗夫格式及其与经典隐式时间步长格式的关系。尽管这类格式在通常的时间步长意义下是稳定的，但将其解释为时空算子方程可能导致条件稳定性，其常数取决于时间和空间网格尺寸之间的关系。我们重新审视阿齐兹和蒙克的连续伽辽金方法，该方法在最低阶情况下得到克兰克 - 尼科尔森格式，并针对高低正则性解提供详细的时空误差分析。特别是，时空框架使我们能够分析经典时间步长方法对于非光滑初始数据的退化行为。通过在时间上应用分部积分，我们推导了一种伴随时空格式，它以自然变分方式纳入初始条件。在最低阶情况下，该格式导致初始数据的兰纳彻型平滑。理论结果通过数值实验得到补充。
英文摘要
We study space-time Galerkin--Petrov formulations for parabolic evolution problems and their relation to classical implicit time-stepping schemes. Although such schemes are stable in the usual time-stepping sense, their interpretation as space-time operator equations may lead to conditional stability, with constants depending on the relation between temporal and spatial mesh sizes. We revisit this phenomenon for the continuous Galerkin method of Aziz and Monk, which yields the Crank--Nicolson scheme in the lowest-order case, and provide a detailed space-time error analysis for solutions of both high and low regularity. In particular, the space-time framework allows us to analyze the deteriorated behaviour of classical time-stepping methods for nonsmooth initial data. By applying integration by parts in time, we derive an adjoint space-time formulation that incorporates the initial condition in a natural variational way. In the lowest-order case, this formulation leads to a Rannacher-type smoothing of the initial data. The theoretical results are complemented by numerical experiments.
