A Mixed Finite Element Method For Sobolev Equation

Sobolev Equation is generally applied in physical problems such as the seepage theory about the fluid crossing cracks in the rock, the migration problem of the moisture in the soil, and the heat-conduction question of different medium. Corresponding numerical method research has presented a large amount of results, (see in [1-5]). In this paper, utilizing a way different from those in the literature, we have done jobs as following:(1) We propose a mixed finite element method responding to Sobolev equation, and construct a lower-order quadrilateral mixed element. By connecting with semi-discrete and fully-discrete schemes given in this work respectively, we also demonstrate the existence, the uniqueness and the error estimation of the discrete solution, and improve the related results in [6]. Numerical experiments show that theory proof and analysis are identical.(2) On the base of (1), we propose corresponding combined finite element methods and analyze the existence, the uniqueness and the error estimation of the finite element solution. Numerical results prove them effectively.(3) Making use of constant flux, we propose corresponding finite element schemes based on the combined partial projection method. Under the circumstance of rectangle space subdivision, numerical experiments prove the effectiveness of this method.