高阶有限体积法中的关键数学问题的研究

The stability (or inf-sup condition) of the FVM is crucial to guarantee the existence and uniqueness of the solution of the FVM and to obtain its error estimate. The stability analysis in the most literatures for high-order FVMs over triangular meshes requires the minimum angle conditions. However, numerical experiments show that the minimum angle condition is not necessary. This project is devoted to studying the stability and the a posteriori error estimate of high-order FVMs. In particular, we will mainly focus on the following four aspects. (1) We shall develop a new analytical method to weaken the minimum angle conditions in the existing high-order FVMs. (2) We shall construct new high-order FVMs over triangular meshes and prove that they have stability independent of the minimum angle conditions. (3) We shall construct high-order FVMs over tetrahedron meshes in three-dimension case and prove that they have stability independent of the mesh conditions. (4) We shall construct simple residual-based a posteriori error estimators and prove their reliability and efficiency. The research aspects of this project are the key mathematical problems in the FVMs. Among the above four aspects, the first、second and third aspects have great academic value in improving the algorithm and the theory of FVMs. The last aspect is the foundation for the design of the adaptive algorithms of high-order FVMs. It will promote the application of high-order FVMs in practical problems.

有限体积法的稳定性(即inf-sup条件)是保证离散问题的解存在唯一以及进行误差估计的基础。对三角网格上的高阶有限体积法，现有文献中的稳定性分析一般需要最小角条件，而数值实验表明最小角条件是不必要的。本项目主要研究高阶有限体积法的稳定性和后验误差估计，具体包括以下四个方面：(1)发展新的分析方法，减弱目前稳定性分析中的最小角条件；(2)构造三角网格上的新型高阶有限体积法，证明其具有不依赖于最小角条件的稳定性；(3)构造三维四面体网格上的高阶有限体积法，证明其具有不依赖于网格条件的稳定性；(4)建立高阶有限体积法的后验误差估计，构造简单易算的残量型后验误差估计子，证明其可靠性和有效性。本项目的研究属于有限体积法中的关键数学问题，研究结果对于完善有限体积法的算法构造及其理论具有重要的学术价值。后验误差估计的研究将为有限体积法的自适应算法设计奠定基础，推进高阶有限体积法在实际问题中的应用。

有限体积法广泛的应用于科学和工程计算，但是其理论发展滞后于其应用，尤其高阶格式的理论 结果还很不完善，成为阻碍有限体积法进一步推广和发展的瓶颈问题。本项目主要研究了：1.求解椭圆问题的任意阶有限体积法的后验误差估计。2.求解非线性椭圆问题的二次有限体积法的先验和后验误差估计。3.求解抛物问题的二次有限体积法的研究。该项目的研究工作一方面完善了二阶有限体积法的理论研究，另一方面高阶有限体积法的后验误差估计的建立为高阶有限体积法的自适应算法设计奠定基础，为高阶有限体积法在流体力学及诸多工程技术领域中的应用提供技术支撑，具有重要的应用价值。在该项目的资助下在国际期刊发表SCI论文3篇，在投稿论文2篇，培养研究生1名。