Abstract
This 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. The 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. This approach includes a small nonlinear system on a coarse grid with size and a linear system on a fine grid with size . It follows from error estimates that asymptotically optimal accuracy can be obtained as long as the mesh sizes satisfy .
1. Introduction
A compressible nuclear waste disposal contamination problem in porous media is presented by the following coupled systems of partial differential equations. The 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 and are the fluid pressure and Darcy velocity, respectively, and is the porosity. is a production term, is a salt dissolution term, is the permeability of the rock, and , the viscosity of the fluid, is dependent upon , the concentration of the brine in the fluid.
Brine:where is the diffusion tensor including the molecular diffusion and mechanical diffusion and , , and . Here, is molecular diffusion. are two direction cosines of Darcy velocity. is an identity matrix.
Heat:where is the temperature of the fluid, , , , , and .
Radionuclide (component ):where is the trace concentration of the th radionuclide, , and
We assume the following:(1)Zero Neumann boundary conditions for the equations(2)The initial conditions are assumed given(3)The medium is vertically homogeneous and take (4)The solutions are smooth and periodic(5) are bounded below by positive constants, and are twice continuously differentiable with bounded partial derivatives about the variables in parentheses(6)
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. The method is used by Bi et al. [5], Chen et al. [6–10], and Liu et al. [11, 12] for solving some nonlinear problem. We shall relegate all of nonlinear iterations on a coarse grid of size much coarser than the original fine grid of size . According to the error estimates in the context, it obtains the asymptotically optimal accuracy to take .
The 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.
2. Notations and Approximation Results
To analyze the temporal discretization on a time interval , let be a positive integer number, , , and . Let denote the usual set of functions with the normwhere if , the integral is replaced by the essential supremum. Let denote the time discrete analogue to the space with the norm
Let and . The weak form is presented as follows:for and .
Assume that is the Raviart–Thomas space of index at least associated with a quasitriangulation of such that the elements have diameters bounded by . Let and be finite-dimensional subspaces of for the approximation of concentrations and temperature, respectively, and we take and as the piecewise-polynomial space of degree at least and , respectively. As in [2, 11, 13], the approximation properties for and are given byfor and , and
If the initial solutions , the characteristics-Galerkin and mixed finite element approximation schemes are to find satisfyingwhere and
Remark. If is located outside , we can join with so that is the outer-normal direction to the boundary at . Take so that , then we define .
In order to deduce the error estimates, we employ the elliptic projections by labeling them with tildes.where and for and introduce the following notations:
Subtracting (19) from (9) and taking , we get the error equation about the pressure function as follows:where and is between and .
Next, combining (20) from (10) at with the test function ,
When , we apply the Taylor expansion and obtain thatwhere is between and .
Combining (32) with (33), we get
Using the deduction as [1, 2], we have
After making the induction hypothesis that , we multiply (36) by and sum over to get
Combining (11) and (21), we get the following equality, in which we choose the test function and sum over :where , and .
Note that
The reminder of the right side items in (38) is just as [2], that is,
It follows from the assumption and Gronwall lemma that
Then, 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:where are satisfied.
Note that the time step is limited to be due to the theoretical proof.
Theorem 1. Define for by system (19)–(23) and assume that the approximation properties (25)–(29) hold. Ifthen there exists a positive constant independent of and , such thatSimilarly, we can get the error estimates of fine grid scheme in norm.
Theorem 2. Define for by system (19)–(23) and assume that the approximation properties (25)–(29) hold. Then, there exists a positive constant independent of , and , such that
3. An Efficient Method
We now use and analyze a two-grid method for iteratively solving the nonlinear problem. The method has two steps. Stage 1. On the coarse grid with a mesh size , solve a small nonlinear system for given by (19)–(23). Stage 2. On the fine grid with a mesh size , solve the following linear system for :whereand the projection on the fine grid is based on the numerical solutions on coarse grid.
The sequential solution processes are defined as follows. Firstly, we apply the Newton iteration to the coupled system on the coarse grid and obtain . Next, combining (48)–(50), we get with and with piecewise linear finite element using a coupled linear system. Finally, from (51) and (52), we can get and by parallel computation.
In order to analyze the linear scheme on the fine grid, we define
According to Taylor expansion, there exists a positive such that
According to (48), (9), and (25),
Like the deduction of (34), we see thatwhere is between and .
Hence, (56) and (57) can give that
The error equation about shows thatwhere , and .
Taking the test function and summing over , we have
Since
It follows from (61)–(63) that
At last, we present the error results of the radionuclide equation and heat equation as follows:
If setting , , and , which accords with the assumption (45), we can get the following theorem of fine grid scheme.
Theorem 3. Define for by system (48)–(52) and assume that the approximation properties (14)–(18) hold. Then, there exists a positive constant independent of , and , such that
Theorem 4. Define for by system (48)–(52) and assume that the approximation properties (14)–(18) hold. Then, there exists a positive constant independent of , and , such that
4. Conclusions and Extensions
The two-grid method presented in this paper reduces the complexity of problem. It involves a small nonlinear system on a coarse grid of size and a linear system on a fine grid of size . It is shown that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation as long as the mesh sizes satisfy in norm. Compared with the implicit scheme, the two-grid method reduces CPU time. Moreover, the method is suitable to make the large-scale computation and long time duration. The future work is to use the discontinuous finite volume element method [14], block-centered finite difference method [15], higher-order finite volume [16], and SAV method [17, 18] to consider this problem.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
The author would like to thank Professor Luping Wang whose comments helped improve this paper.