Sobolev方程H^1—Galerkin扩展混合有限元方法

H^1 - GALERKIN EXPANDED MIXED FINITE ELEMENT METHOD FOR LINEAR SOBOLEV EQUATION

为克服H^1-Galerkin混合有限元方法在数值模拟具小扩散系数或低渗透率问题时，因对扩散系数求逆带来的困难，基于H^1-Galerkin与扩展混合有限元的思想，对刻画扩散、渗透过程的Sobolev问题建立了H^1-Galerkin扩展混合有限元格式，证明了格式的稳定性和收敛性质．论证表明该格式具有无需对小扩散系数求逆，较好地克服了小扩散系数带来的困难；能同时高精度逼近未知函数，梯度及其通量，有限元空间无需满足LBB条件；刚度矩阵对称正定等H^1-Galerkin方法和扩展混合有限元法的良好性质．数值算例说明了所提算法的有效性．

To overcome the difficulties arising from calcualtiong the inverse of a small diffusive coefficient when simulating the diffusive problems within a low permeability zone by a mixed finite element method, we develop a H^1 - Galerkin expanded mixed finite method for sobolev problems which govern the diffusive phenomena, and prove the stability and convergence for the proposed procedure. The analysis also shows that the method inherits the advantages of H^1 - Galerkin and expanded mixed finite element methods, such as approximating the unknown, its gradient and its flux directly; the finite element space being free of LBB condition as well as the stiff matrix being symmetry and positive-definite. Numerical tests are performed to confirm the theoretical analysis.