对流反应扩散方程的稳定化时间间断时空有限元解的误差估计

THE ERROR ESTIMATES OF THE STABILIZED TIME DISCONTINUOUS SPACE-TIME FINITE ELEMENT SOLUTIONS FOR CONVECTION-REACTION-DIFFUSION EQUATIONS

在流线迎风Petrov-Galerkin(SUPG)稳定化有限元数值格式的基础上,结合时间方向的变分离散,构造对流反应扩散方程的稳定化时间间断时空有限元格式.该类格式在工程上有一些数值模拟应用,但相关文献没有看到类似数值格式的理论证明.本文以Radau点为节点,构造时间方向的Lagrange插值多项式,证明了稳定化有限元解的稳定性,时间最大模、空间L2(Ω)-模误差估计.文中利用插值多项式和有限元方法相结合的技巧,解耦时空变量,去掉了时空网格的限制条件,提供了时间间断稳定化时空有限元方法的理论证明思路,克服了因时空变量统一导致的实际计算时的复杂性.

A kind of stabilized time discontinuous space-time finite element formulations are presented for the convection-reaction-diffusion equations.The stabilized schemes considered in this paper are constructed based on the streamline upwind Petrov-Galerkin(SUPG)finite element approximate scheme,combining with the space time finite element method in which the discrete time variation form was used.The theoretical proofs of the scheme cannot be found in relative references,although there are some simulations in practical applications.In this paper,we give the proofs of stability and error estimates in the maximum norm of time and L2(Ω)-norm of spatial by taking the Radau points as the interpolation nodes of Lagrange interpolation polynomials.The temporal and spacial variants are decoupled,the restricted conditions on the space-time meshes are removed by taking advantages of the techniques of combining the interpolation polynomials with the finite element method.The process of of proof in this paper provides a new idea of theoretical analysis for the stabilized space time finite element method permitting discontinuities in time.And the complexities in practical computations caused by the unifying space and time variables are conquered.