The presence of corners in the computational domain, in general, reduces the regularity of solutions of parabolic problems and diminishes the convergence properties of the finite element approximation introducing a so-called "pollution effect". Standard remedies based on mesh refinement around the singular corner result in very restrictive stability requirements on the time-step size when explicit time integration is applied. In this article, we introduce and analyse the energy-corrected finite element method for parabolic problems, which works on quasi-uniform meshes, and, based on it, create fast explicit time discretisation. We illustrate these results with extensive numerical investigations not only confirming the theoretical results but also showing the flexibility of the method, which can be applied in the presence of multiple singular corners and a three-dimensional setting. We also propose a fast explicit time-stepping scheme based on a piecewise cubic energy-corrected discretisation in space completed with mass-lumping techniques and numerically verify its efficiency.
High-Order Multirate Explicit Time-Stepping Schemes for the Baroclinic-Barotropic Split Dynamics in Primitive Equations
为了有效处理海洋动力学的多时间尺度问题，斜压-正压分裂技术被广泛应用于海洋模拟原始方程的求解。基于保持强稳定性的Runge-Kutta方法，本文提出了两种高阶多速率显式时间步格式（SSPRK 2-SE和SSPRK 3-SE）。所提出的格式允许大的时间步长用于三维斜压（慢）模式和小的时间步长用于二维正压（快）模式，其中两种模式的求解只需要满足各自的CFL条件以保证数值稳定性。在每个时间步，先用大时间步\textcolor{black}{and the assistance of some intermediate approximations of the baroctropic mode obtained by substepping with the small-time step}推进系统的斜压模式和流体厚度，计算斜压速度，然后用小时间步再推进正压模式，利用前一个斜压模式解的强迫项插值得到改进的正压强迫，对正压速度进行修正;最后，通过斜压和正压速度的耦合更新流体厚度。此外，数值不一致的离散海面高度所造成的模式分裂缓解通过和解过程仔细计算通量赤字。在“MPAS-Ocean”平台上进行了两个基准测试，数值验证了所提出的SSPRK-SE方案的性能和并行可扩展性。
Comparing EPGP Surrogates and Finite Elements Under Degree-of-Freedom Parity
Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, USA（数学与统计学系，旧 Dominion 大学）；Department of Electrical Engineering and Computer Science, OWL University of Applied Sciences and Arts（电气工程与计算机科学系，OWL 应用科学与艺术大学）；Department of Mathematics and Statistics, Georgetown University, Washington, DC 20057, USA（数学与统计学系，杰克逊维尔大学）
A New $L2-1_σ$-Interior Penalty Method for Variable-Order Time-Fractional Subdiffusion Interface Problem with Curved Interface
School of Arts and Sciences, Guangzhou Maritime University（广州航海学院人文与科学学部）；School of Information Engineering, Zhejiang Ocean University（浙江海洋大学信息工程学院）；Fuzhou Vocational Technical College（抚州职业技术学院）
