The L2-Projection and Quasi-Optimality of Galerkin Methods for Parabolic Equations
The L2-Projection and Quasi-Optimality of Galerkin Methods for Parabolic Equations
复制标题
抛物方程伽辽金法的L2投影和拟最优性
DOI：
--
复制
发表时间：
2016
影响因子：
2.9
通讯作者：
A. Veeser
中科院分区：
文献类型：
--
作者：
F. Tantardini;A. Veeser
我们考虑线性抛物型初边值问题，并分析了空间上的Galerkin逼近。借助于inf-sup理论，我们得到了关于标准弱公式和只需要时间可积性的公式所产生的范数的准最优性结果。此外，我们还揭示了$L^2~-投影的$H^1~-稳定性不仅是充分的，而且是这些结果的必要条件。作为应用，我们考虑了空间中的协调有限元逼近，并根据局部网格尺寸和分段正则性得到了先验误差界。该正则性是逼近理论所示的最小正则性，与线性抛物问题的正则性结果相吻合。
We consider linear parabolic initial-boundary value problems and analyze Galerkin approximation in space. With the help of the inf-sup theory, we derive quasi-optimality results with respect to norms that arise from the standard weak formulation and from a formulation requiring only integrability in time. Moreover, we reveal that the $H^1$-stability of the $L^2$-projection is not only sufficient but also necessary for these results. As application, we consider conforming finite element approximation in space and derive a priori error bounds in terms of the local meshsize and piecewise regularity. The regularity is the minimal one indicated by approximation theory and matches regularity results for linear parabolic problems.