在本研究中，我们提出了一种真正的四阶紧致有限差分格式，用于在二维和三维中求解具有Dirichlet边界条件的双调和方程。在二维情形下，我们在Zhilin Li和Kejia Pan最初为通量型边界条件开发的高阶紧致（HOC）格式的基础上[SIAM J. Sci. Comput., 45 (2023), pp. A646-A674]，构造了耦合边界条件的高阶紧致离散。在考虑三维情形时，我们修改了Li和Pan精心设计的待定系数方法，以推导耦合边界条件的有限差分近似。所得的有限差分离散保持了全局四阶收敛性和紧致性。与非常流行的Stephenson方法不同，未知数的数量不随维数增加。此外，值得注意的是，在二维和三维中，系数矩阵的条件数均以$O(h^{-2})$的速率增长。我们还通过非平凡算例验证了所提出的真正HOC方法的性能。
英文摘要：
In this study, we propose a genuine fourth-order compact finite difference scheme for solving biharmonic equations with Dirichlet boundary conditions in both two and three dimensions. In the 2D case, we build upon the high-order compact (HOC) schemes for flux-type boundary conditions originally developed by Zhilin Li and Kejia Pan [SIAM J. Sci. Comput., 45 (2023), pp. A646-A674] to construct a high order compact discretization for coupled boundary conditions. When considering the 3D case, we modify carefully designed undetermined coefficient methods of Li and Pan to derive the finite difference approximations of coupled boundary conditions. The resultant FD discretization maintains the global fourth order convergence and compactness. Unlike the very popular Stephenson method, the number of unknows do not increase with dimensions. Besides, it is noteworthy that the condition number of the coefficient matrix increases at a rate of $O(h^{-2})$ in both 2D and 3D. We also validate the performance of the proposed genuine HOC methods through nontrivial examples.
On $C^{0}$ Interior Penalty Method for Fourth Order Dirichlet Boundary Control Problem and a New Error Analysis for Fourth Order Elliptic Equation with Cahn-Hilliard Boundary Condition
Polar differentiation matrices for the Laplace equation in the disk subjected to nonhomogeneous Dirichlet, Neumann and Robin boundary conditions and the biharmonic equation subjected to nonhomogeneous Dirichlet conditions
The fully nonconforming virtual element method for biharmonic problems
Politecnico di Milano（米兰理工大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Istituto di Matematica Applicata e Tecnologie Informatiche - CNR（意大利国家研究委员会应用数学与信息技术研究所）
Explicit block-encodings for biharmonic boundary-value problems
School of Mathematical Sciences, East China Normal University（华东师范大学数学科学学院）；Key Laboratory of MEA, Ministry of Education, East China Normal University（华东师范大学教育部MEA重点实验室）；Shanghai Key Laboratory of PMMP, East China Normal University（华东师范大学上海PMMP重点实验室）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
Existence results and numerical solution for the Dirichlet problem for fully fourth order nonlinear equation
Center for Informatics and Computing, Vietnam Academy of Science and Technology(VAST)（越南科学技术院信息计算中心）；College of Sciences, Thainguyen University（太原大学理学院）
A Matrix-free Augmented High Order Compact Solver for Variable-Coefficient Biharmonic Problems
College of Science, National University of Defense Technology（国防科技大学理学院）；School of Mathematics and Statistics, HNP-LAMA, Central South University（中南大学数学与统计学院）；Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University（南洋理工大学物理与数学科学学院数学科学系）
Conforming and non-conforming virtual element methods for the biharmonic Steklov eigenvalue problem with minimum regularity
Indian Institute of Technology-Kharagpur（印度理工学院卡拉格普尔分校）；King Abdullah University of Science and Technology (KAUST)（阿卜杜拉国王科技大学）；Università di Pavia（帕维亚大学）；IMATI, Consiglio Nazionale delle Ricerche（国家研究委员会IMATI研究所）；Universidad Técnica Federico Santa María（费德里科·圣塔玛丽亚技术大学）
