双调和方程的高阶非协调有限体积法

The finite volume method (FVM) is one of the most commonly used numerical methods for solving partial differential equations. It has been widely used in the industry and fluid computation. However, the fact that the theoretical analysis of FVMs lags far back their applications seriously hinders the further promotion and development of the methods. The algorithm construction and theoretical analysis of FVMs, especially higher-order schemes, is a difficult task ungent to be solved and it attracts many computational mathematicians to focus on this field. The main purpose of this project is to present a general setting for the construction of test spaces in the case that the trial spaces for FVMs are assumed to be standard nonconforming finite element spaces and to provide the unified analysis for the FVMs constructed here. The general construction given here will cover the existing specific FVM schemes. More importantly, it will provide many new interesting FVM schemes. The uniform boundedness and uniform local-ellipticity of the family of the discrete bilinear forms lead to the H1 error estimate of FVMs. We will establish the uniform boundedness of the family of the discrete bilinear forms with the help of the equivalent discrete norms. The algebraic and geometric sufficient and necessary conditions for the uniform local-ellipticity will be established and convenient sufficient conditions for the uniform local-ellipticity will also be derived.

有限体积法是求解偏微分方程的一种重要的数值解法，在工业界和流体计算中得到广泛的实际应用。但是，其理论研究相对滞后于方法的应用，严重阻碍了该方法的进一步推广和发展，成为了有限体积法研究者所面临的最为迫切的问题。特别地，高阶有限体积法格式的构造和分析一直是计算数学界的研究热点。该项目致力于研究高阶非协调有限体积法的一般性的算法构造和统一的理论分析框架。对于前者，我们拟对高阶非协调有限体积法建立一般性的构造框架，使该框架覆盖并发展已有的算法构造成果；对于后者，我们考虑利用等价的离散范数导出离散双线性型族的一致有界性，然后建立离散双线性型族一致局部椭圆性的充分必要条件以及方便使用的充分条件，从而得到有限体积法的H1误差估计

最近三十年，有限体积法已经广泛地应用到了计算流体动力学、计算力学等工程领域。然而，高阶格式，尤其是高阶非协调元格式理论的滞后，严重阻碍了该方法的进一步推广和发展。本项目对二阶椭圆方程和四阶双调和方程的高阶非协调有限体积法进行了研究。对二阶椭圆方程非协调有限体积法，基于三角剖分和矩形剖分的情况我们已经取得了预期的成果。对双调和方程的高阶非协调有限体积法，我们对几种已有的具体的格式逐一分析，得到最优误差估计，并且构造了几种新的高阶非协调有限体积法格式，理论结果也得到最优。但是，我们希望能给出一个统一的算法和理论构造框架，这个工作还在进行中。这些结果对于建立和完善有限体积法的一般构造和理论分析具有相当重要的学术价值。