High Accuracy Schemes And Rapid Algorithms For Some Kinds Of Partial Differential Equations With Nonlocal Boundary Conditions

The initial-boundary value problems for parabolic equations and their inverse parabolic problems with nonlocal boundary conditions,and Laplace-Beltrami operator eigenvalue problem with nonlocal boundary conditions have been widely applied in various fields of science and engineering,such as thermoelasticity,heat conduction,image processing and so on.At present,although the researches of numerical methods,theoretical analysis and applications of these problems have made great progress,there are still many problems to be further studied.In this work,four classes of PDE with nonlocal boundary are studied,and the major research findings and innovations are described as follows.For a one dimensional parabolic equation with an integral two-space-variable condition,we first build a backward Euler scheme.Then,we introduce some new methods and techniques on basis of the discrete Fourier transform(DFT),and prove that under a general condition ? ? Ch2 where C is a positive constant independent of mesh size,the errors of the scheme at boundary points and interior points can reach the optimum asymptotic orders O(?|ln h|)and O(?ln2 h)in the maximum norm,respectively.Furthermore,we present two formulas to approximate partial derivatives of the exact solution,and prove that the approximation formula for ut has the super approximations of O(?(1 + ?/h? |ln h|)and O(?(1 +?/h)ln2 h)at the boundary point and the interior point,respectively.In addition,we also prove that the approximation formula for ux2 has the super approximation of O(? ln2 h)at interior points which keep a certain distance with boundaries.Finally,numerical experiments are presented to demonstrate the validity of the theoretical results.For a two-dimensional parabolic equation with nonlocal boundary conditions,we first build a backward Euler difference scheme.Then,based on the eigenfunction corresponding to the initial boundary value problem of parabolic equation,a new transform is constructed,and combined it with the discrete Fourier transform the three-dimensional error analysis problem is ingeniously transformed into one dimen-sion problem;On this basis,we prove that the errors of the scheme at interior points can reach the optimum asymptotic order O((? + h2)|ln h|)in the maximum norm.Furthermore,we present formulas to approximate two spatial partial derivations ux and uy of the exact solution,and prove that the approximation formulas for ux and uy have the super approximations of O((?+h2)|In h|)and O((?+h2)ln2 h)at the interior points,respectively.In the end,numerical experiments are presented to demonstrate the validity of the theoretical results.For a one-dimensional parabolic inverse problem with an unknown time-dependent function in boundary conditions,we first build a backward Euler difference scheme and prove that the scheme can reach the optimum asymptotic order at boundary points and interior points are O(?|ln h|)and O(?ln2 h)in the maximum norm,re-spectively.Next,we present a formula to approximate the spatial partial derivative ux of the exact solution and prove that this formula has the super approximations of O(? ln2 h)at interior points.Moreover,we present a approximation formula for the unknown function ?(t)and prove it has the super approximations of O(? ln2 h)at all temporal points.In the end,numerical experiments are presented to demonstrate the validity of the theoretical results.After building a linear FEM scheme for two dimensional Laplace-Beltrami oper-ator eigenvalue problem with periodic boundary condition,which is derived from the image segmentation background,we investigate that how the distance between target object and boundary affects the segmentation performance obtained by the eigenfunc-tion corresponding to the minimum eigenvalue on a single layer grid.Then,based on the adaptive biscetion coarsening strategy,we desgin a new fast two-grid algorithm to solve the discrete system of eigenvalue problems.In the end,numerical experiments are presented to verify the efficiency and the robustness of the new algorithm.