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Discrete Maximal Regularity for the Discontinuous Galerkin Time-Stepping Method without Logarithmic Factor
SIAM Journal on Numerical Analysis ( IF 2.8 ) Pub Date : 2024-07-22 , DOI: 10.1137/23m1580802 Takahito Kashiwabara 1 , Tomoya Kemmochi 2
SIAM Journal on Numerical Analysis ( IF 2.8 ) Pub Date : 2024-07-22 , DOI: 10.1137/23m1580802 Takahito Kashiwabara 1 , Tomoya Kemmochi 2
Affiliation
SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1638-1659, August 2024.
Abstract. Maximal regularity is a kind of a priori estimate for parabolic-type equations, and it plays an important role in the theory of nonlinear differential equations. The aim of this paper is to investigate the temporally discrete counterpart of maximal regularity for the discontinuous Galerkin (DG) time-stepping method. We will establish such an estimate without logarithmic factor over a quasi-uniform temporal mesh. To show the main result, we introduce the temporally regularized Green’s function and then reduce the discrete maximal regularity to a weighted error estimate for its DG approximation. Our results would be useful for investigation of DG approximation of nonlinear parabolic problems.
中文翻译：
无对数因子的间断伽辽金时间步进法的离散最大正则性
《SIAM 数值分析杂志》，第 62 卷，第 4 期，第 1638-1659 页，2024 年 8 月。
抽象的。最大正则性是抛物型方程的一种先验估计，在非线性微分方程理论中发挥着重要作用。本文的目的是研究不连续伽辽金（DG）时间步进方法的最大正则性的时间离散对应物。我们将在准均匀时间网格上建立这样一个没有对数因子的估计。为了显示主要结果，我们引入了时间正则化格林函数，然后将离散最大正则性降低为其 DG 近似的加权误差估计。我们的结果对于研究非线性抛物线问题的 DG 近似很有用。
更新日期：2024-07-22
Abstract. Maximal regularity is a kind of a priori estimate for parabolic-type equations, and it plays an important role in the theory of nonlinear differential equations. The aim of this paper is to investigate the temporally discrete counterpart of maximal regularity for the discontinuous Galerkin (DG) time-stepping method. We will establish such an estimate without logarithmic factor over a quasi-uniform temporal mesh. To show the main result, we introduce the temporally regularized Green’s function and then reduce the discrete maximal regularity to a weighted error estimate for its DG approximation. Our results would be useful for investigation of DG approximation of nonlinear parabolic problems.
中文翻译：
无对数因子的间断伽辽金时间步进法的离散最大正则性
《SIAM 数值分析杂志》，第 62 卷，第 4 期，第 1638-1659 页，2024 年 8 月。
抽象的。最大正则性是抛物型方程的一种先验估计，在非线性微分方程理论中发挥着重要作用。本文的目的是研究不连续伽辽金（DG）时间步进方法的最大正则性的时间离散对应物。我们将在准均匀时间网格上建立这样一个没有对数因子的估计。为了显示主要结果，我们引入了时间正则化格林函数，然后将离散最大正则性降低为其 DG 近似的加权误差估计。我们的结果对于研究非线性抛物线问题的 DG 近似很有用。
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