数学物理中的总体和局部场GL方法

为了求解物理化学生物材料和金融中的微分方程,提出了一种总体(Global)和局部(Local)场方法.微分方程的求解区域可以是有限域,无限域,或具曲面边界的部分无限域.其无限域包括有限有界不均匀介质区域.其不均匀介质区域被分划为若干子区域之和.在这含非均匀介质的无限区域,将微分方程的解显式地表示为在若干非均匀介质子区域上和局部子曲面的积分的递归和.把正反算的非线性关系递归地显式化.在无限均匀区域,微分方程的解析解被称为初始总体场.微分方程解的总体场相继地被各个非均匀介质子区域的局部散射场所修正.这种修正过程是一个子域接着另个子域逐步相继地进行的.一旦所有非均匀介质子区域被散射扫描和有限步更新过程全部完成后,微分方程的解就获得了.称其为总体和局部场的方法,简称为GL方法.GL方法完全地不同于有限元及有限差方法,GL方法直接地逐子域地组装逆矩阵而获得解.GL方法无需求解大型矩阵方程,它克服了有限元大型矩阵解的困难.用有限元及有限差方法求解无限域上的微分方程时,人为边界及其上的吸收边界条件是必需的和困难的,人为边界上的吸收边界条件的不精确的反射会降低解的精确度和毁坏反算过程.GL方法又克服了有限元和有限差方法的人为边界的困难.GL方法既不需要任何人为边界又不需要任何吸收边界条件就可以子域接子域逐步精确地求解无限域上的微分方程.有限元和有限差方法都仅仅是数值的方法,GL方法将解析解和数值方法相容地结合起来.提出和证明了三角的格林函数积分方程公式.证明了当子域的直经趋于零时,波动方程的GL方法的数值解收敛于精确解.GL方法解波动方程的误差估计也获得了.求解椭圆型,抛物线型,双曲线型方程的GL模拟计算结果显示出我们的GL方法具有准确,快速,稳定的许多优点.GL方法可以是有网,无网和半网算法.GL方法可广泛应用在三维电磁场,三维弹塑性力学场,地震波场,声波场,流场,量子场等方面.上述三维电磁场等应用领域的GL方法的软件已经由作者研制和发展了。

A New GL Method for Mathematical and Physical Problem

We propose a new GL method for solving the ordinary and the partial differential equation.These equations govern the electromagnetic field etc.macro and micro physical,chemical,financial problems in the sciences and engineering.The domain can be finite,infinite,or part of the infinite domain with a curve surface.The differential equation is held in an infinite domain which includes a finite inhomogeneous domain.The inhomogeneous domain is divided into finite sub domains.We present the solution of the differential equation as an explicit recursive sum of the integrals in the inhomogeneous sub domains.We discover the explicit relationship between forward modeling and inversion.The analytical solution of the equation in the infinite homogeneous domain is called as an initial global field.The global field is updated by local scattering field successively subdomain by subdomain.Once all subdomains are scattered and the finite updating process is finished in all the sub domains,the solution of the equation is obtained.We call our method as Global and Local field method,in short GL method.The GL method is totally different from Finite Element Method(FEM) method and Finite Difference Method(FD),the GL method directly assemble inverse matrix and get solution successively subdomain by subdomain.There is no big matrix equation needs to solve in the GL method which overcome FEM' and FD's difficult for solving big matrix equation.When the FEM and FD are used to solve the differential equation in the infinite domain,the artificial boundary and absorption boundary condition are necessary and difficult.The error reflections from the artificial absorption boundary condition downgrade the accuracy of the forward solution and damage the inversion resolution.The GL method resolves the artificial boundary difficulty in FEM and FD methods.There is no artificial boundary and no absorption boundary condition for infinite domain in the GL method.We proposed a triangle integral equation of the Green's functions and proved several theorems for the theoretical analysis of the GL method.The numerical discretization of the GL method is presented.We proved that the numerical solution of the GL method is convergent to the exact solution when the maximum diameter of the sub domain is going to zero.The error estimation of the GL method for solving the wave equation is presented.The simulations show that the GL method is accurate,fast,and stable for solving elliptic,parabolic,and hyperbolic equations.The GL method has wide applications in the 3D electromagnetic (EM) field,3D elastic and plastic etc seismic field,acoustic field,flow field,and quantum field.The GL method software for the above 3D EM etc field is developed by authors in GL Geophysical Laboratory.