论文标题
四面体网格的二次有限体积方法的构建和分析
Construction and analysis of the quadratic finite volume methods on tetrahedral meshes
论文作者
论文摘要
在四面体网格上构建和分析了二次有限体积法（FVM）方案的家族。为了证明稳定性和误差估计，我们提出了四面体网格上的最小V角条件，以及双网格上的表面和音量正交条件。通过元素分析技术，局部稳定性等同于$ 9 \ times9 $元素矩阵的正确定性，这很难直接或数值分析。借助表面正交条件和一致转换，该元素矩阵被降低到块对角矩阵中，然后在最小V-Engle条件下执行稳定性结果。值得一提的是，四面体外壳的最小V-ENGLE条件与三角形网格的最小角度条件的简单扩展非常不同，而在实践中也很方便。基于稳定性，我们证明了最佳$ H^{1} $和$ l^2 $错误估计，其中正交条件在确保最佳$ L^2 $收敛率方面起着重要作用。提出了数值实验，以说明我们的理论结果。
A family of quadratic finite volume method (FVM) schemes are constructed and analyzed over tetrahedral meshes. In order to prove stability and error estimate, we propose the minimum V-angle condition on tetrahedral meshes, and the surface and volume orthogonal conditions on dual meshes. Through the element analysis technique, the local stability is equivalent to a positive definiteness of a $9\times9$ element matrix, which is difficult to analyze directly or even numerically. With the help of the surface orthogonal condition and congruent transformation, this element matrix is reduced into a block diagonal matrix, then we carry out the stability result under the minimum V-angle condition. It is worth mentioning that the minimum V-angle condition of the tetrahedral case is very different from a simple extension of the minimum angle condition for triangular meshes, while it is also convenient to use in practice. Based on the stability, we prove the optimal $ H^{1} $ and $L^2$ error estimates respectively, where the orthogonal conditions play an important role in ensuring optimal $L^2$ convergence rate. Numerical experiments are presented to illustrate our theoretical results.
