Stokes-Darcy方程有限元方法保多项式梯度重构  

Polynomial-preserving recovery finite element method for the Stokes-Darcy system

作　　者：崔明[1] 王亚楠[1] 张百驹 张智民 Ming Cui;Yanan Wang;Baiju Zhang;Zhimin Zhang

机构地区：[1]北京工业大学理学部,北京100124 [2]云南大学数学与统计学院,昆明650500 [3]Department of Mathematics,Wayne State University,Detroit,MI 48202,USA

出　　处：《中国科学：数学》2024年第8期1105-1122,共18页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:12131005);北京市自然科学基金(批准号:1192003)资助项目。

摘　　要：近年来,对于Stokes-Darcy方程的研究受到越来越多的关注.一方面是因为它在水文学、环境科学和石油工程等诸多领域有着广泛的应用;另一方面,该方程在数学理论和数值分析方面也十分具有挑战性.保多项式重构(polynomial-preserving recovery,PPR)方法是一种后处理方法,将其用于有限元的后处理,可以获得连续更优的梯度逼近结果.本文对Stokes-Darcy方程的有限元方法给出基于PPR的超收敛分析和后验误差估计.利用有限元数值结果和PPR方法,本文重构网格节点处的梯度近似值,证明重构后的梯度超收敛于精确梯度,并证明后验误差估计在渐近意义上是精确的.数值实验验证了本文的理论分析结果.In recent years,the Stokes-Darcy system has attracted much attention due to its wide applications in hydrology,environmental science,petroleum engineering,and many other fields.In this paper,we study fi-nite element methods for the Stokes-Darcy system,especially the superconvergence property and the a posteriori error estimation based on the polynomial-preserving recovery(PPR).PPR is a post-processing technique that reconstructs numerical gradients with higher accuracy.We prove that the gradient reconstructed by PPR is su-perconvergent,and the a posteriori error estimator based on PPR is asymptotically exact.Numerical experiments are presented to validate our theoretical analysis.

关 键 词：Stokes-Darcy方程 保多项式重构 后验误差估计 超收敛 

分 类 号：O241.82[理学—计算数学]