四阶椭圆型偏微分方程及相关问题渐近保结构的降阶计算方法研究

Fourth-order elliptic partial differential equations and related problems form a big research field of mathematics and applied sciences with wide sources and applications; the numerical methods of this filed possess significant importance and big development space. At the same time, abundant results on algorithms, theories and softwares have accumulated in the field of numerical methods for partial differential equations, and it is thus of nonnegligible significance to have the existing results reused for higher efficiency and benefits in the development of the field in future. Based on these observations, the project proposes to study the design, analysis and implementation of the multi-level adaptive finite element methods of fourth-order elliptic partial differential equations and related problems, with the order-reduced method which is the bonding point of the two trends being a main tool. In the project, three kinds of equations with the backgrounds each from multi disciplines are used as model problems, which are the biharmonic equation in two and three dimension, the quad curl equations in three dimension and the vector Laplacian equation in three dimension; topics relevant to them will be studied. The idea of asymptotically structure-preserving is used to balance the necessity of structure preservation for the need of stability and accuracy and the induced difficulties in designing discretizations. The proposed project will also study how to implement the finite element software platform PHG on the implementation of the proposed algorithms as well as how to enlarge the functions and applications of the PHG motivated by new problems. This can be viewed as an initiatory exploration on the interaction between computational methods of new kinds and standardized software packages.

四阶椭圆型偏微分方程及相关问题是数学和应用科学中具有丰富来源和广泛应用的研究领域，其数值方法有显著的研究意义和发展空间。与此同时，偏微分方程数值方法领域已经积累了丰富的算法、理论和软件成果，重复利用已有的成果提高研究开发的效率和效益在该领域未来发展中有不可忽视的意义。本项目以上述两个发展方向的结合点——降阶方法为主要手段，以具有多学科应用背景的二维和三维双调和方程、三维四阶旋度方程、三维向量Laplace方程等为典型模型问题，通过渐近保结构思想来平衡满足稳定性和精度要求与克服此要求带来的离散方法设计上的困难这两个方面，研究四阶椭圆型方程及相关问题的多水平自适应有限元方法的设计、分析与实现。本项目研究内容包括利用并行自适应有限元软件平台PHG实现所设计的数值方法和利用新问题为PHG平台增加新功能模块并扩大其应用范围，初步探索新型计算方法与标准化软件互相支持互相带动的发展模式。

四阶椭圆型偏微分方程及相关问题是数学和应用科学中具有丰富来源和广泛应用的研究领域，其数值方法有显著的研究意义和发展空间，是本项目的整体的出发点和驱动力。以四阶问题的元形式和降阶形式的等价关系及其在离散层面上的保持为中心问题，本项目围绕一些重要的四阶模型问题或模型问题的组合及其变体问题展开研究，取得了一定的研究成果。包括：1）系统性的研究了四阶问题的降阶格式的构造，为若干四阶算子的边值问题和特征值问题发展了基于低正则性有限元空间的离散格式，并建立了一般性的具有渐进保结构特征的离散格式构造框架；2）比较系统的发展了一类非Ciarlet型的有限元格式，构造了一些在Ciarlet有限元范畴内不容易构造起来的有限元格式；3）构造了一些具有临界特征的有限元格式，回答了关于临界格式的存在与否的问题，发现了一些非标准的数值现象。本项目的新成果、新问题、新方法观念有望对应用学科和下一步的研究产生直接影响。