The Research On Numerical Solution For Eliptic Equation And Its Stability Analysis

Addressed in this dissertation is the numerical method of elliptic equation and itsstability analysis. Several relevant problems attracting more and more attention in thisfield are discussed in detail, and a series of well-established, systematical, and importantresults are obtained.The backgrounds and research status on the boundary integral equation and ellipticequation are introduced. It also provides the readers with some preliminaries and impor-tant definitions and lemmas that are frequently used in the remaining chapters.Mechanical quadrature methods and splitting extrapolation are studied for solvingthe first kind of boundary integral equations with singularities. On the basis of the mid-point rule and Sidi-Israeli’s quadrature formulas for the boundary integral equations withweak singularities, we can derive formulas for one dimension Cauchy singular integraland their corresponding Euler-Maclaurin expansion. We also propose the quadrature for-mulas for the Two-dimension singular integral on the basis of quadrature formulas for theone dimension Cauchy singular integral, and construct the mechanical quadrature meth-ods. The existence and uniqueness of numerical solutions are studied by spectral analysis.The convergence and stability of numerical solutions are obtained through the collectivelycompact theory. The multi-parameter asymptotic expansion of errors is provided and thealgorithm of splitting extrapolation method is presented too. By splitting extrapolationmethod we can not only get the high accuracy of approximations, but also derived a pos-teriori error estimate for adaptive algorithms.The finite difference method is considered for solving biharmonic equations. Hirtmethod is applied to judge the stability for the multi-level difference schemes of the bi-harmonic equation. The existence of finite difference method solution is discussed byestimating the lower bound of the minimum eigenvalue for the discrete matrix. Moreover,using Taylor series expansion the convergence of finite difference method solution is re-searched. In addition, the asymptotic expansions of the errors are given and the accuracyorder of approximation solutions is improved by extrapolation, which is verified by ex-periments.Nextly, Stability analysis is investigated by the finite difference method for bihar- monic equations. Firstly, the effective condition number is applied to the standard 13-point finite difference equation for biharmonic equations, and the bounds of effectivecondition number are derived. It is proved that the bounds of the effective condition num-ber are O(h^-3.5) in general cases, which are smaller than Cond. = O(h^-4). Surprisingly,the bounds of the effective condition number are only O（1） for homogeneous boundaryconditions. Namely, these new stability analysis is more valid than previous stability anal-ysis. Numerical experiments are provided to verify the stability analysis. Secondly, withthe help of a transformation v = u - uˉ（uˉsatisfies the non-homogeneous boundary con-ditions） the biharmonic equation with non-homogeneous boundary conditions convert toa new biharmonic equation with homogeneous boundary conditions, which has a smallercondition number. Hence an excellent stability for numerical biharmonic equations canbe achieved.A method is proposed for estimating lower bounds for the minimum eigenvalues ofmatrix, which uses the character of special matrices to obtain the lower bounds. Althoughthe method is not perfect, it is very useful in practical applications. Numerical examplesconfirm the theoretical results.