The Convergence Analysis Of A Nonconforming Finite Element With The Application In Numerical Reservoir Simultion

Miscible displacement of one incompressible fluid by another in a porous media in numerical reservoir simulation can be described by the mathematical models: Equation (0.1) is called the pressure equation; equation (0.2) is called the concentration equation. The unknown in the model is the Darcy velocity u, pressure p, and concentration c. In numerical reservoir simulation, we main care about the concentration c; because concentration c is determined by the Darcy velocity u. if we first solve the pressure equation for pressure and then calculate Darcy velocity by the gradient of the pressure, and finally calculate the concentration by the concentration equation, this will decrease the accu-racy of the approximated concentration. The mixed finite element method allows us to calculate the Darcy velocity and pressure at the same time, which will increase the accuracy of the approximated concentration. In this article we use the mixed finite element method to solve the pressure equation.In conventional finite element method, we usually assume regular assump-tion or quasi-uniform assumption, namely hk/ρk≤M or hmax/hmin≤M, hk represent the max diameter of unit K. ρk is the max diameter of the Inscribed circle of unit K. Jh is a split family of domain. However, with the development of the finite element method, the above assumptions have more and more deficient, this article studied the miscible incompressible displacement in porous media based on the research of non-conforming finite element by Shi Dongyang. We used two methods to study this model; The first method:we use nonconforming finite element to approximate the pressure equation and use the conforming finite clement to approximate the concentration equation:the second method: we use the nonconforming finite element to approximate both the pressure equation and the concentration equation.The article is divided into four chapters:The first chapter briefly introduces the porous media incompressible mis-cible displacement model, as well as the corresponding variational form.Chapter two briefly introduces the limitations of the regular assumption or quasi-uniform assumption, mainly introduce the construct and the nature of the nonconforming finite element.Chapter three introduce the first method of this article, namely using the nonconforming finite element to approximate the pressure equation and using the conforming finite element to approximate the concentration equation, and derive the following error estimate:Chapter four introduce the second method of this article, namely using the nonconforming finite element to approximate both the pressure equation and the concentration equation, and derive the following error estimate:...