解Stokes方程的一种无散稳定二次有限体积法格式  

A DIVERGENCE-FREE STABLE QUADRATIC FINITE VOLUME METHOD SCHEME FOR THE STOKES EQUATIONS

作　　者：张杰华 韩明华 Zhang Jiehua;Han Minghua(Department of Mathematics,College of Science,Kaili University,Guizhou 556011,China)

机构地区：[1]凯里学院理学院,凯里556011

出　　处：《计算数学》2024年第1期79-98,共20页Mathematica Numerica Sinica

基　　金：凯里学院基金(BS201710);贵州省教育厅基金([2018]361);国家自然科学基金(11961038)资助。

摘　　要：在三角形网格上构造了一种求解Stokes方程的Lagrange二次有限体积法格式.取连续的二次有限元空间与间断的线性有限元空间分别作为Stokes方程的速度项与压力项的试探空间,从而保证了离散方程的速度解在宏元三角形单元上满足局部质量守恒性,且有限元空间对自然满足所谓的inf-sup条件.采用特殊的有限体积法映射与对偶剖分,求解Stokes方程的Lagrange二次有限体积法格式等价于相对应的有限元法格式,因此确保了有限体积法格式的无条件(无需约束三角形网格的几何形状)稳定性和关于速度项的最优阶H1范数的误差估计.最后,数值实验展示了理论结果的正确性以及有限体积法的数值模拟在计算流体力学中的有效性.A Lagrange quadratic finite volume method scheme for solving the Stokes equation is constructed on triangular meshes in this paper.The piecewise continuous quadratic finite element space and the discontinuous linear finite element space is used as the trial space for velocity and pressure of the Stokes equation respectively,so that the discrete velocity solution of the finite volume method satisfies the local mass conservation on the macroelement triangular element,and the finite element space pair is naturally satisfied with the so-called inf-sup condition.By adopting the special dual partition and the special mapping,the finite volume method scheme for solving the Stokes equation is transformed into the corresponding finite element method.The unconditional stability(or inf-sup condition)of the finite volume method scheme(without the geometric constraints of the triangular meshes)and the optimal-order error estimates in the H~1-norm for velocity are obtained.Finally,numerical experiments show the validity of the theoretical results and the effectiveness of the finite volume method in the numerical simulation of computational fluid dynamics.

关 键 词：有限体积法 STOKES方程 无散 稳定性 

分 类 号：O241.82[理学—计算数学]