多孔介质驱动问题的混合元最小二乘特征有限元方法及其收敛分析

１．引言多孔介质二相驱动问题的数学模型是偶合的非线性偏微分方程组的初边值问题．该问题可转化为压力方程和浓度方程［１－４］．浓度方程一般是对流占优的对流扩散方程，它的对流速度依赖于比浓度方程的扩散系数大得多的Ｆａｒｃｙ速度．因此Ｄａｒｃｙ速度的求解精度直接影响着浓度的求解精度．为了提高速度的求解精度，７０年代Ｐ．Ａ．Ｒａｖｉａｔ和Ｊ．Ｍ．Ｔｈｏｍａｓ提出混合有限元方法［５］．Ｊ．ＤｏｕｇｌａｓＪｒ，Ｔ．Ｆ．Ｒｕｓｓｅｌｌ，Ｒ．Ｅ．Ｅｗｉｎｇ，Ｍ．Ｆ．Ｗｈｅｅｌｅｒ［１］－［４］，［９］，［１２］袁…

THE MIXED LEAST-SQUARE AND CHARACTERISTIC FINITE ELEMENT METHODS FOR TWO-PHASE DISPLACEMENT IN POROUS MEDIA

The mathematical model for two-phase displacement in porous media is a coupled initial boundary value problem of nonlinear partial differential equations which consist of a pressure equation and a saturation equation. In this paper, the mixed least-square weak form of pressure equation is got, and the positive definite characteristics of the weak form is proved. Based on this weak form, a new kind of numerical methods for two-phase displacement problems is proposed: the mixed least-square finite element method is used to solve pressure and Darcy velocity, and the saturation is solved by using characteristic finite element method. The main merits of the mixed least-square finite element method compared with mixed finite element method are: first, the structure of the mixed least-square finite el- ement spaces is just standard finite element spaces, it is simple and easy to use; second, the weak form of the mixed least-square finite element method for pressure is symetric and definite positive, thus there are many efficient methods to solve numerically; and the last, the Darcy velocity solved by mixed least-square finite element method is continuous. In numerical analyses, a very important inequality is obtained which is used to control the errors of the pressure and Darcy velocity, and the optimal error estomates of the proposed method are proved.