Numerical Solutions And Preconditioning Techniques For Several Kinds Of Differential/Integral Equations

In many fields, such as electromagnetic, fluid mechanics, and thermonuclear reaction, differential(or integral) equations play an important role to understand the behind mechanism of evolution, transformation, reformation, etc. These are highly complicated PDEs and an explicit and simple analytic solution is often unavailable. Hence, in general numerical computation is the unique tool that we can rely on to study these PDEs. Numerical methods for these PDEs usually contain two stages: in the first stage we discretize the PDEs and in the second stage we solve the resulting algebraic equations directly or iteratively. For the first stage, some typical methods, such as the finite difference methods(FDMs), the finite element methods(FEMs), the finite volume methods(FVMs) and the spectral methods, have been extensively studied over the past years. However, to match the increasing requirement of speed, accuracy and memory cost for solving these PDEs,it is desirable to design efficient numerical methods to realize fast computation with less memory cost but high accuracy. For solving the obtained algebraic equations, especially the large sparse linear systems, the Krylov subspace methods are often an attractive family of choices. These iterative methods are popular for their robust performances and low memory requirements. Nevertheless, as we know, the convergence rates of almost all subspace iterative methods are deeply affected by the distributions of eigenvalues or singular values of the corresponding coefficient matrix. Consequently, to establish efficient preconditioners for the targeted algebraic equations plays a vital role in the whole numerical simulations. This thesis aims at systematically solving several partial differential equations(PDEs) and an integral equation. The main contents are organized as follows:1. By splitting the singular integral operator, we present a collocation method based on the piecewise Lagrange interpolation function for the airfoil integral equation. Then,using an approximate inverse of the singular operator, we introduce a preconditioner.Numerical experiments are carried out to show the effectiveness of the proposed preconditioner even when the nonsingular integral term is perturbed.2. Using the quadratic spline function, we solve the Helmholtz equation subject to Sommerfeld boundary conditions. Numerical examples show that this method can well approximate the unknown function and its first, second order derivatives. By reordering the mesh points, we transform the coefficient matrix of the corresponding linear system into a 3 × 3 block matrix, where the three diagonal elements are all block tridiagonal matrices. Then, with the aid of the stair matrix, we construct a block polynomial preconditioner. Numerical examples illustrate the proposed preconditioner is better than the conventional ILU preconditioner as the mesh points are increased.3. Based on the quadratic spline function, a collocation method is introduced to solve the time fractional subdiffusion equation, then, we analyze the accuracy of this method and the characteristics of the obtained coefficient matrix. Being different from the traditional FDM, we find that the accuracy of this collocation method is not affected by the order β. In fact, this method possesses a global error bound O(Ï„3+ h3) and can achieve the accuracy of O(Ï„4+ h4) at all collocation points, where Ï„, h are the step sizes in time and space respectively. For the obtained linear system, we present a preconditioner by employing the idea of splitting operator, numerical examples show this preconditioner performs well for the small order β.4. For the generalized saddle point problems arising from solving the Navier-Stokes equation with FEM, we introduce a splitting preconditioner, and implement it via using the Sherman-Morrison-Woodbury formula. Theoretical analyses and examples are reported to show that this preconditioner is efficient.5. When numericallly solving optimization, eigenvalues problems and nonlinear PDEs, one usually gets the shifted linear system(A + α I)x = b. Researching preconditioning techniques of the shifted system will be conducive to fast solving these problems.Based on the ILU factorization of the seed matrix A, we introduce a preconditioning strategy by adjusting the non-zero elements of L,U. This strategy still remains the sparsity of the original factorization. Theoretical analyses show that when α â†' 0, ∞, the eigenvalues of the preconditioned matrix will cluster around 1. Some numerical examples are given to verify this conclusion.