Convergence Analysis for Mixed Finite Element Method of Positive Semi-definite Problems

A mixed element-characteristic finite element method is put forward to approximate three-dimensional incompressible miscible positive semi-definite displacement problems in porous media. The mathematical model is formulated by a nonlinear partial differential system. The flow equation is approximated by a mixed element scheme, and the pressure and Darcy velocity are computed at the same time. The concentration equation is treated by the method of characteristic finite element, where the convection term is discretized along the characteristics and the diffusion term is computed by the scheme of finite element. The method of characteristics can confirm strong computation stability at the sharp fronts and avoid numerical dispersion and nonphysical oscillation. Furthermore, a large step is adopted while small time truncation error and high order accuracy are obtained. It is an important feature in numerical simulation of seepage mechanics that the mixed volume element can compute the pressure and Darcy velocity simultaneously and the accuracy of Darcy velocity is improved one order. Using the form of variation, energy method, L2 projection and the technique of priori estimates of differential equations, we show convergence analysis for positive semi-definite problems. Then a powerful tool is given to solve international famous problems.


Introduction
The incompressible miscible positive semi-definite displacement problem in porous media consists of two partial different equations: an elliptic equation for the pressure, a convection-diffusion equation for the concentration, where the concentration equation has strong hyperbolic feature (Douglas, 1983;Dougals, Ewing & Wheeler, 1983;Ewing, Russell & Wheeler, 1984;Russell, 1985), Ω denotes a bounded domain of R 3 , and ν is the normal outer vector to the boundary surface, denoted by ∂Ω.The pressure, p(X, t), Darcy velocity, u = (u 1 , u 2 , u 3 ) T and the concentration of water, c(X, t), are objective functions.q(X, t), the quantity, is greater than zero at injection wells and is less than zero at production wells, and q = max{q, 0}.ϕ(X) is the porosity of rock, and κ(X) is absolute permeability.µ(c), the viscosity of mixture, depends on c. c, the concentration of injected fluid, is equal to c at production wells, c = c.γ(c) and d(X) = (0, 0, z) T denote the gravitational coefficient and vertical coordinates, respectively.The diffusion matrix, D(X, u), is generally defined by (Yuan & Han, 2008;Yuan, Wang & Han, 2010), x ûx ûy ûx ûz ûx ûy û2 y ûy ûz ûx ûz ûy ûz û2 D m is the molecular diffusivity.I denotes a 3×3 unit matrix.α l and α t are the longitudinal and the transverse dispersivities, respectively.ûx , ûy , ûz denote three direction cosines of u.Generally speaking, the symbol, β ≥ 2, is a positive constant.The mathematical model is usually used to describe numerical simulation of oil reservoir and contaminant-transport problem, and the diffusion matrix is supposed to be positive definite.While in actual numerical simulation applications such as oil-gas resources basin assessment (Yuan & Han, 2008;Yuan, Wang & Han, 2010) and numerical computation of enhanced (chemical) oil recovery (Yuan, Cheng, Yang & Li, 2014,2015), the diffusion matrix is only positive semi-definite (Dawson, Russell & Wheeler, 1989;Ewing, 1983;Yuan, 2013), The present paper mainly considers a positive semi-definite problem, and the discussion gives more theoretical reference in terms of mathematics and mechanics (Ewing, 1983;Yuan, 2013).
Oil-water two phase seepage displacement is a primary topic in numerical simulation of oil reservoir.For two-dimensional positive definite problems, Douglas and Russell presented well-known numerical methods such as characteristic finite difference and characteristic finite element (Russell, 1985;Douglas, 1983).Douglas, Ewing and Wheeler put forward the method of mixed element (Douglas, Ewing & Wheeler, 1983), and Ewing, Russell, Wheeler discussed the characteristicsmixed element (Ewing, Russell & Wheeler, 1984).The above arguments were based on the positive definite assumption, but the diffusion matrix was only positive semi-definite in some actual applications (Dawson, Russell & Wheeler, 1989;Ewing, 1983;Yuan & Han, 2008;Yuan, Wang & Han, 2010;Yuan, 2013;Yuan, Cheng, Yang & Li, 2014,2015).Therefore, the framework of theoretical analysis is not feasible.It is hard and difficult to show convergence analysis of semidefinite problem.The characteristic finite element method was presented by Dawson (Dawson, Russell & Wheeler, 1989).For three-dimensional positive semi-definite problems, Yuan discussed characteristic finite element and characteristic finite difference (Yuan, 1997(Yuan, ,1999)).Based on the above discussions, we present a method of mixed element-characteristic finite element to simulate three-dimensional incompressible miscible positive semi-definite displacement problem of (1)-(4).The flow equation is treated by a conservative mixed element method, and the pressure and Darcy velocity are obtained at the same time.The characteristic finite element is used to solve the concentration equation, where the convection term is discretized along the characteristics and the diffusion term is approximated by the finite element method.The characteristics can confirm strong stability at the fronts and can avoid numerical dispersion.Moreover, it can adopt large spatial step while computational accuracy is not decreased.More important in numerical simulation of seepage mechanics, the pressure and Darcy velocity are obtained simultaneously by using the scheme of mixed element and the computational accuracy of Darcy velocity is developed one order.Using variation form, energy method, L 2 projection and theoretical framework of priori estimate, we show convergence analysis in L 2 norm.Then the well-known difficult problem is solved numerically, and a basic theoretical reference is given for actual numerical simulations.
common symbols and notations of Sobolev space are adopted.Suppose that the problem of ( 1)-( 4) is regular, (R) where l ≥ 3, k ≥ 1.
And suppose that the problem is positive semi-definite where a * , a * , ϕ * and ϕ * are positive constants.
In this paper, the symbols K and ε denote a generic positive constant and a generic small positive number, respectively.They have different definitions at different places.

The Mixed Element for the Pressure
The form of variation is discussed.Let H(div; Ω) denote a space consisting of vector functions, v ∈ L 2 (Ω) 3 , satisfying The pressure p(X, t) is determined only except an additive constant.For simplicity, consider a factor space For α, β ∈ V, φ ∈ W and θ ∈ L ∞ (Ω), define the following bilinear function Then the pressure equation is equivalent to a family of saddle point problems: The problem of ( 11) is considered.Let h p > 0 be the spatial step for the pressure, and let J h p be a quasi-regular partition of Ω, consisting of tetrahedrons or cubes with the greatest diameter at most h p .Let V h × W h ⊂ V × W be a Raviar-Thomas space on the partition (Raviart & Thomase, 1977;Thomase, 1977), with the index k and the approximation O(h k+1 p ), whose approximations satisfy Introduce elliptic projection of (u, p) to find ( ũh , ph ): where c is the exact concentration.
It is seen that in the references (Douglas, Ewing & Wheeler, 1983;Ewing, Russell & Wheeler, 1984) the solution ( ũh , ph ) exists solely and is estimated as follows Then it follows from ( 14) and The mixed element scheme is constructed.When the approximate concentration It has been proved that numerical solutions of ( 16) exist solely (Brezzi, 1974).Using ( 14) and ( 15), The concentration equation ( 2) is discretized later.

The Finite Element Approximation for the Concentration
For convenience to interpret the approximation of concentration, we suppose that Darcy velocity u = (u 1 , u 2 , u 3 ) T is given.
The procedures are constructed by combining the discretization of characteristics and the approximation of finite element.
Let M h ⊂ H 1 (Ω) be a normal finite element space with index l, and let h c > 0 be a spatial step of quasi-regular partition J h c .The largest diameter of tetrahedron elements or cube elements is not exceeding h c .Approximation order is O(h l+1 c ) (Ciarlet, 1978), Let τ(X, t) denote a unit vector of the characteristics, and let Then the characteristic derivative is formulated by Let ∆t c = T/N denote the time step for the concentration, and The characteristic derivative, ∂c n ∂τ (X) = ∂c ∂τ (X, t n ), is approximated by a backward difference quotient where The variation of ( 2) is defined as follows.A function, c : J → H 1 (Ω), is determined by By using ( 19), ( 22) is restated by Since the diffusion matrix D(u) is positive semi-definite, so L 2 projection is introduced, replacing elliptic projection, to show convergence analysis.For t ∈ J, ch ∈ M h is defined by Then the estimate holds (Ciarlet,1978) The characteristic finite element scheme of ( 23) is constructed.
where c0 h is an L 2 projection of initial solution c 0 (X).

The Composite Scheme
Combining ( 16) and ( 26), we state a composite scheme to solve (1)-( 4).In actual computation Darcy velocity changes more slowly than the saturation with respect to time t, so spatial large step is adopted for computing (16).Time interval J is partitioned 0 All the steps except for the first step ∆t 1 p are supposed to be uniform ∆t m p = ∆t p , m ≥ 2. Each pressure node t m is also a saturation node t n where m, n are positive integers, and let j = ∆t p /∆t c , j 1 = ∆t 1 p /∆t c .For a function φ m (X) = φ(X, t m ) related with saturation step t n for t m−1 < t n ≤ t m , we require a velocity approximation u h in (26).If m ≥ 2, define a linear extrapolation of u h,m−1 and u h,m−2 as follows 16) with ( 26), replacing exact solution by numerical approximations, then we can obtain full discrete coupled scheme of ( 1)-( 4) to find where ĉn−1 ).The procedures of ( 28) and ( 29) run as follows.
Step 4. Similarly, we get the values of Step 5.The program runs repeatedly as above, then all the numerical solutions are obtained.

Convergence Analysis
Based on the discussions of Darcy velocity u h and the pressure p h , ( 14) and ( 17), convergence analysis is shown as follows.Let ζ = c h − ch and ξ = c − ch .From (28a), (23a) (t = t n ), ( 21) and ( 24), taking z h = ζ n , we have where The second term of (31), 1 and let Estimate the other terms on the left-hand side of ( 30), The right-hand side of ( 30) is considered as follows, An induction hypothesis is used sup 0≤n≤L−1 The last term is discussed.