Domain Decomposition Preconditioner And Its Application In Numenrical Simulation Of Groundwater

With the in-depth study of the numerical simulation of groundwater, researchers make higher requirements for real-time, accurate, detailed and credible information of groundwater simulation, and have pressing needs for the numerical simulation software that can efficiently solve problems with the characteristics of fine meshes and long time span.The fine subdivision of the study area often leads to a higher memory and lower efficiency problem. In order to solve the mentioned problem, on the one hand, researchers select new discretization methods for the mathematical model, such as (multi-scale finite element method, Laplace transform finite layer method, and the FAC method, etc.) to reduce the equations dimension and memory storage with less meshes, and to improve the efficiency under certain precision. On the other hand, reaearchers pay much attention on efficient algorithms for solving the large, sparse, morbid linear algebraic equations. And so far the preconditioned conjugate gradient method (PCG) has already been an extremely efficient algorithm for solving large sparse linear algebraic equations. Furthermore, the construction of efficient preprocessor is the key for PCG.In recent years, domain decomposition method is popular because of its unique advantages. In the paper, the preprocessor is built with domain decomposition methods and detailed implementation processes of the domain decomposition preprocessor (DDP) are given. Domain decomposition preconditioned conjugate gradient method (DDP-PCG) is made with the combination of DDP and PCG method. A confined groundwater problem with the analytical solution is used to verify the credibility of DDP-PCG. Under different discretizations of the study area,CG Jacobi-PCG, SSOR-PCG, DDP-PCG methods are used respectivly to solve the above-mentioned problem. Results show that the number of iterations of DDP-PCG are significantly lower than other methods in a variety of grid scale, and almost independent of the grid scale indicating the robustness of DDP-PCG. However, the CPU time-consuming of DDP-PCG is not the lowest which is caused by the low efficiency for solving the sub-region problems.A fast algorithm for the sub-region problems is the cornerstone of the domain decomposition algorithm. Based on the finite element theory, fourier analysis method(FAM) is studied for the homogeneous confined steady flow problem in a rectangular area with11kinds of combination of the Dirichlet and Neumann boundary conditions. A variety of transformation formulas in the FAM are calculated using the FFT algorithm and the corresponding computer program are prepared to realize the fast solution for the homogeneous confined steady flow problems. Then, the results of FAM are compared to the analytical solution or other numerical solution for11numerical experiments to indicate the credibility of FAM. And the numerical experiment for the confined steady flow model with the single well of fiexed pumping flow rate which has the analytic solution further validates the credibility of FAM. Under different subdivision of the study area, FAM and iterative methods (Jacobi, Gauss-Seidel, SOR, PCG) are used to solve the above problem. And calculation results show that:under a certain precision, the finer the discretization of the study aera is, the more significant is the efficiency of FAM. In addition, for the homogeneous confined steady flow problem, FAM does not need the calculation and storage of the original coefficient matrix, which makes much savement of the memory space.When the subregion problems of DDP-PCG are calculated with FAM, we get the domain decomposition preconditioned conjugate gradient method based on fourier analysis(FA-DDP-PCG). As for hydrogeological problems, the sub-region division modes, corresponding FAM for the subregions, as well as the solving steps of FA-DDP-PCG are given and corresponding computer program are prepared. FA-DDP-PCG is applied for homogeneous confined flow model with the analytical solution to verify the reliability of FA-DDP-PCG. Then CG, Jacobi-PCG, SSOR-PCG, FA-DDP-PCG are used to solve the model under different subdivisions, which show that FA-DDP-PCG is of higher efficiency compared to other methods and the finer the subdivision is, the more notable is FA-DDP-PCG’s efficiency under a certain precesion. Finally, confined aquifer with continuous coefficients and abrupt coefficients are caculated under finely subdivision, which further proves that FA-DDP-PCG’s efficiency are the highest of all. For the case with continuous coefficients, randomized trials explaine that akij (which represents the permeability coefficient or transmissibility) has little effect on the algorithm’s efficiency, which indicates FA-DDP-PCG is of strong robustness. For the case with abrupt coefficients, experiments with different parameters are caculated, which show that FA-DDP-PCG can effectively solve groundwater model with strongly abrupt coefficients. In all, FA-DDP-PCG is an effective method for solving large-scale groundwater flow problems.