多边形／多面体网格上的有限元方法

Recently, numerical methods constructed on polygonal/polyhedral meshes for solving partial differential equations have attracted considerable attention due to their flexibility in dealing with complicated geometry, flexibility in adaptive mesh refinements, as well as their increased accuracy in approximating the curved boundary. Among the various new methods, we propose to study the conforming finite element method defined on polygonal/polyhedral meshes, because of both its theoretical significance and practical straightforwardness. There are a vast amount of blanks in this new research area to be filled by researchers. The proposed project will focus on: (1) studying the polygonal/polyhedral mesh regularity, mesh quality, and how the finite element approximation error depends on the mesh quality; (2) constructing and analyzing higher order H1 conforming finite elements on polygonal/polyhedral meshes; (3) constructing and analyzing H(curl) and H(div) conforming mixed finite elements satisfying the discrete de Rham sequence; (4) constructing efficient multigrid preconditioners; (5) studying the conforming finite element discretization of elasticity problems, plate bending problems, and the Stokes equations on polygonal/polyhedral meshes.

近年来，在多边形／多面体网格上求解偏微分方程的数值方法，由于其在处理复杂区域和自适应网格时的灵活性，以及对曲边／曲面逼近精度的提高，而受到极大关注。这些新方法中，基于广义重心坐标的协调有限元方法，以其理论上的重要性与实现上的直观简便，成为数值分析中不可或缺的一环，这也将是我们要研究的内容。这个新兴方向目前存在大量空白亟待填补，本项目将重点研究其中几个问题：（1）多边形／多面体网格的正则性、网格质量、以及有限元误差与网格质量的依赖关系；（2）构造并分析多边形／多面体网格上的高阶H1协调有限元；（3）构造并分析满足离散de Rham sequence的H(curl)与H(div)混合有限元；（4）构造高效的多重网格预处理算法；（5）研究弹性问题、板问题、Stokes方程在多边形／多面体网格上的协调有限元离散。

多边形/多面体网格上的偏微分方程数值离散，由于其在处理复杂区域和自适应网格时的灵活性，近年来受到大量关注。本项目主要考虑多边形/多面体网格上基于广义重心坐标的有限元方法，主要成果如下：提出了多边形/多面体网格上各向异性的网格质量参数，并基于此实现了多边形移动网格算法；构造了多边形与部分多面体网格上满足离散de Rham sequence的H(curl)与H(div)协调有限元，给出误差分析；利用广义重心坐标构造并分析了多边形网格上的Crouzeix-Raviart型非协调有限元；构造了多边形网格上Stokes方程LBB稳定的Bernardi-Raugel型离散；构造了一种多边形网格上基于广义重心坐标的H1协调二次元，并利用该元来构造了Stokes方程LBB稳定的二次元离散；研究了一类四阶定常与非定常问题的Ciarlet-Raviart混合有限元离散，并应用于拟线性的薄膜外延生长问题模拟；研究了广义重心坐标有限元在平面弹性问题离散中的应用。