Sobolev方程的H^1-Galerkin时空混合有限元分裂格式

H^1-Galerkin space time mixed finite element splitting scheme for one-dimensional Sobolev equation

研究了一维Sobolev方程的H^1-Galerkin时空混合有限元分裂格式,格式中有限元空间可以利用不同次数的多项式空间,不需要满足LBB条件,避免求解耦合方程组,能同时得到时间和空间两个变量的形式高阶精度,且能同时高精度逼近渗透流体的浓度u,浓度梯度q和流体通量σ.通过格式分裂,时空统一处理,引入时空投影算子等方法,证明了H^1-Galerkin时空混合有限元解的存在唯一性,稳定性和误差估计,并给出数值算例验证格式的有效性和可行性以及理论分析结果的合理性.

A kind of space time mixed finite element splitting scheme for one dimensional Sobolev equation is constructed.The scheme has some advantages,for example it can use different polynomial space as finite element space,it need not LBB condition,it can avoid solving coupled equations,the scheme can obtain high order accuracy both in space and time,and it can simultaneously approximate the concentration of the permeating fluid u,the concentration gradient q and the fluid fluxσ.The stability and existence as well as uniqueness,error estimates of the H^1-Galerkin space time mixed finite element solution are derived by the technique of splitting scheme,unifying space and time variables,introducing space and time projections.Finally,numerical examples are given to verify the validity and feasibility of the format and the rationality of the theoretical analysis results.