对流反应扩散方程的SUPG稳定化时空有限元解的误差估计

Error Estimations of SUPG Stabilized Space-Time Finite Element Approximations for Convection-Diffusion-Reaction Equations

将时空有限元方法和流线扩散迎风Petrov-Galerkin方法(SUPG)相结合,构造对流扩散反应方程的一种全离散稳定化时空有限元方法.和传统的SUPG方法不同,本文为得到高精度尤其是时间高精度格式,在时空两个方向同时使用离散变分形式.该类格式曾被工程师用来数值模拟一些实际问题,但很难看到相关文献的理论分析证明.本文时间方向利用Gauss-Legendre和Gauss-Lobatto积分,并和有限元方法相结合,证明数值解的稳定性和误差估计.不但去掉时空网格的限制条件,而且将时间和空间变量解耦,克服了时空有限元方法在建立格式时由于时空变量统一处理而导致的理论分析和数值模拟中的高维度难度和复杂性,本文不需要引入对偶问题的证明思路丰富了稳定化SUPG时空有限元方法的理论.

A kind of fully discrete version of stabilized space-time finite element approximate scheme for the convection-diffusion-reaction equations is established by combining the space-time finite element discretization with the streamline upwind Petrov-Galerkin method(SUPG).The formulations discussed in this paper is different from traditional SUPG method.The discrete variation forms were used in both time and space directions in order to derive high order accuracy schemes,especially high accuracy in time.The theoretical proofs of this kind of scheme were difficult to find in relative references,although there are some simulations in practical applications studied by engineers.We focus on presenting the proof of stability and error estimates of the approximate solutions.The techniques of combining the Gauss-Legendre and Gauss-Lobatto integration rules with the finite element method was used.The conditions on the space-time meshes are removed,the space and time variables are decoupled.This kind of decoupling can overcome the high dimensional difficulties and complexities in both theoretical analysis and practical computations caused by unifying the space and time variables when established the scheme.And the idea of the proof without introducing dual problem presented here will build up the theoretical foundations of the stabilized SUPG space-time finite element scheme.