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.