A Numerical Method for Compressible Model of Contamination from Nuclear Waste in Porous Media

-is paper studies and analyzes a model describing the flow of contaminated brines through the porous media under severe thermal conditions caused by the radioactive contaminants. -e problem is approximated based on combining the mixed finite element method with the modified method of characteristics. In order to solve the resulting algebraic nonlinear equations efficiently, a two-grid method is presented and discussed in this paper. -is approach includes a small nonlinear system on a coarse grid with size H and a linear system on a fine grid with size h. It follows from error estimates that asymptotically optimal accuracy can be obtained as long as the mesh sizes satisfy H � O(h1/3).


Introduction
A compressible nuclear waste disposal contamination problem in porous media is presented by the following coupled systems of partial differential equations. e physical processes can be concreted to be a high-level waste disposal buried in a salt dome, and next the salt dissolves to generate a brine, radioactive elements decay to generate heat, and finally the radionuclides are transported by the flow. Fluid: where p and u are the fluid pressure and Darcy velocity, respectively, ϕ 1 � ϕc w and ϕ is the porosity. q � q(x, t) is a production term, R s ′ (c) � [c s ϕK s f s /(1 + c s )](1 − c) is a salt dissolution term, k(x) is the permeability of the rock, and μ(c), the viscosity of the fluid, is dependent upon c, the concentration of the brine in the fluid. Brine: where E c is the diffusion tensor including the molecular diffusion and mechanical diffusion and E c � D + D m I, D � (D) i,j � ((α T |u|δ i,j + (α L − α T )u i u j )/|u|), and g(c) � − c [c s ϕK s f s /(1 + c s )](1 − c) − q c + R s . Here, D m is molecular diffusion. u i and u j are two direction cosines of Darcy velocity. I is an identity matrix. Heat: where c i is the trace concentration of the i th radionuclide, We assume the following: (1) Zero Neumann boundary conditions for the equations (2) e initial conditions are assumed given (3) e medium Ω is vertically homogeneous and take Ω ∈ R 2 (4) e solutions are smooth and Ω periodic (5) k, ϕ, ϕ 1 , d 2 , and K i are bounded below by positive constants, and , and Q(T) are twice continuously differentiable with bounded partial derivatives about the variables in parentheses (6) D � 0 Chou and Li [1], Ewing et al. [2], and Li et al. [3] have presented and studied several numerical methods for system (1)-(5) and its incompressible case. In this paper, we use the mixed finite element method to approximate the fluid problem and treat the brine, heat, and radionuclides by a modified method of characteristic finite element. It is well known that the full discrete approximation scheme is coupled and nonlinear. If simply lagging, the evaluation of the nonlinear items is used to obtain a linear system; it would be inevitable to introduce the constraint conditions about the mesh grid due to the stability requirement. Moreover, it would take an expensive cost to choose the implicit scheme to nonlinear solutions. An efficient method motivated by Xu [4] is considered in this paper. e method is used by Bi et al. [5], Chen et al. [6][7][8][9][10], and Liu et al. [11,12] for solving some nonlinear problem. We shall relegate all of nonlinear iterations on a coarse grid of size H much coarser than the original fine grid of size h. According to the error estimates in the context, it obtains the asymptotically optimal accuracy to take H � O(h 1/3 ). e remainder of the paper is organized as follows. Notations and approximation assumptions are given in Section 2. A two-grid method is defined and the convergence error estimates are derived in Section 3. In Section 4, we give some conclusions and extensions.

Notations and Approximation Results
To analyze the temporal discretization on a time interval (0, T), let M be a positive integer number, △t � T/M, t n � n△t (0 ≤ n ≤ M), and φ n � φ(·, t n ). Let L p (J; W j,q (Ω)) denote the usual set of functions with the norm where if p � ∞, the integral is replaced by the essential supremum. Let l p (J; W j,q (Ω)) denote the time discrete analogue to the space L p (J; W j,q (Ω)) with the norm ; v · c � 0 . e weak form is presented as follows: for z ∈ H 1 (Ω) and i � 1, . . . , N.
Assume that V h × W h is the Raviart-omas space of index at least k associated with a quasitriangulation of Ω such that the elements have diameters bounded by h p . Let for the approximation of concentrations and temperature, respectively, and we take M h and R h as the piecewise-polynomial space of degree at least l and r, respectively. As in [2,11,13], the approximation properties for V h × W h and M h , R h are given by inf for inf If the initial solutions h , the characteristics-Galerkin and mixed finite element approximation schemes are to find where i � 1, . . . , N and Remark. If x n− 1 is located outside Ω, we can join In order to deduce the error estimates, we employ the elliptic projections by labeling them with tildes.
where U, P : J ⟶ V h × W h , C: J ⟶ M h , T: J ⟶ R h and C i : J ⟶ M h for t ∈ J and introduce the following notations: Subtracting (19) from (9) and taking w � d t α n− 1 , we get the error equation about the pressure function as follows: Mathematical Problems in Engineering where d t α n− 1 � (α n − α n− 1 )/Δt and δ n 1 is between c n and c n h . Next, combining (20) from (10) at t � t n with the test function β n , When t � t n− 1 , we apply the Taylor expansion and obtain that where δ 2 is between c and c h . Combining (32) with (33), we get By (31) and (34), Using the deduction as [1,2], we have (36) After making the induction hypothesis that sup ‖β n ‖ ∞ ⟶ 0, we multiply (36) by Δt and sum over Combining (11) and (21), we get the following equality, in which we choose the test function z � χ n − χ n− 1 � d t χ n− 1 Δt and sum over 1 ≤ n ≤ M: where e reminder of the right side items in (38) is just as [2], that is, It follows from the assumption sup‖∇χ n− 1 ‖ ∞ ≤ C and Gronwall lemma that en, by the inverse estimate and (40), we know that the induction hypotheses hold if Finally, from the approximation properties, Similar to the above analysis, we can obtain the error estimates for the radionuclide equation and heat equation as follows: then there exists a positive constant C independent of h and △t, such that

An Efficient Method
We now use and analyze a two-grid method for iteratively solving the nonlinear problem. e method has two steps.
where Mathematical Problems in Engineering