# GL method for solving equations in math-physics and engineering

Article

## Abstract

In this paper, we propose a GL method for solving the ordinary and the partial differential equation in mathematical physics and chemics and engineering. These equations govern the acustic, heat, electromagnetic, elastic, plastic, flow, and quantum etc. macro and micro wave field in time domain and frequency domain. The space domain of the differential equation is 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. Actualy, we propose an explicit representation of the inhomogeneous parameter nonlinear 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 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. It is different from FEM method, the GL method directly assemble inverse matrix and gets solution. There is no big matrix equation needs to solve in the GL method. There is no needed artificial boundary and no absorption boundary condition for infinite domain in the GL method. We proved several theorems on relationships between the field solution and Green’s function that is the theoretical base of our GL method. The numerical discretization of the GL method is presented. We proved that the numerical solution of the GL method convergence to the exact solution when the size of the sub domain is going to zero. The error estimation of the GL method for solving 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 advantages and 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 are developed.

## Keywords

GL method electromagnetic acoustic nanometer materials dispersion analytical and numerical Quantum

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## References

1. 
Brandts, J., Krizek, M. History and future of superconvergence in three dimensional finite element methods. Mathematical Sciences and Applications, 15: 24–35 (2001)
2. 
Cui, J.Z. Variational method and finite element method for plane stress analysis. Mathematical Practice and Knowledge, (1): 23–34 (1972)Google Scholar
3. 
Feng, D., Jin, G.J. Condensed physics. Shanghai Scientific Technology Press, 1992 (Chinese)Google Scholar
4. 
Feng, K. Difference scheme based on the variational principle. Applied Mathematics and Computational Mathematics, (4): 238–262 (1966)Google Scholar
5. 
Huang, H.C., Wang, J.X., Cui, J.Z., Zhao, J.F., Lin, Z.K. Difference scheme of the variational principle in the elastic mechanics. Applied Mathematics and Computational Mathematics, (3): 54–60 (1966)Google Scholar
6. 
Krizek, M. Superconvergence phenomena on three dimenional meshes. International journal of numerical analysis and modeling, 2(1): 43–56 (2005)
7. 
Li, J.H., Xie, G.Q., Xie, L., Xie, F. A 3D GL EM modeling and inversion for forest exploration and felling. PIERS Online, 3(4): 402–410 (2007)
8. 
Li, J.H., Xie, G., Xie, L. A new GL method for solving differential equation in electromagnetic and physchemical and financial mathematics. PIERS Online, 3(8): 1151–1159 (2007)
9. 
Li, J.H., Xie, G., Oristaglio, M., Xie, L., Xie, F. A 3D-2D AGILD EM modelin and inversion imaging. PIERS Online, 3(4): 423–429 (2007)
10. 
Li, J., Xie, G., Xie, F. New stochastic AGLID EM modeling and inversion. PIERS Online, 2(5): 490–494 (2006)
11. 
Li, J., Xie, G., Li, J. 3D and 2.5D AGLID EMS stirring modeling in the cylindrical coordinate system. PIERS Online, 2(5): 505–509 (2006)
12. 
Li, J.H., Xie, G.Q., Xie, L., Xie, F. GL time domain modeling for EM acoustic and elastic wave field with dispersion in crystal and porous material. Proceeding of PIERS, 2008, p.655–662Google Scholar
13. 
Lin, Q., Yan, N.N. The construction and analysis of high effective finite element methods. Hebei University Press Ltd, 1996Google Scholar
14. 
Xie, G.Q. A new iterative method for solving the coefficient inverse problem of wave equation. Communication on Pure and Applied Math., 39: 307–322 (1986)
15. 
Xie, G.Q. Three dimensional finite element method for solving the elastic problem. Mathematical Practice and Knowledge, (1): 28–41 (1975)Google Scholar
16. 
Xie, G.Q., Chen, Y.M. A modified pulse spectrum technique for solving inverse problem of two-dimensional elastic wave equation. Appl. Numer. Math., 1(3): 217–237 (1985)
17. 
Xie, G.Q., Li, J.H. Nonlinear integral equation of coefficient inversion of acoustic wave equation and TCCR iteration. Science In China, 32(5): 513–523 (1989)
18. 
Xie, G.Q., Li, J.H. New parallel GILD-SOR modeling and inversion for E-O-A strategic simulation: IMACS series book in Computational and Applied Math., 1999, Vol. 5. p.123–138Google Scholar
19. 
Xie, G.Q., Li, J.H. New parallel SGILD modeling and inversion. Physics D, 133: 477–487 (1999)
20. 
Xie, G.Q., Li, J.H. A new algorithm for 3-D nonlinear electromagnetic inversion. 3-D Electromagnetic Methods, SEG book, 1999, Vol.7: 193–207. Editor: Oristaglio and Spies.Google Scholar
21. 
Xie, G.Q., Li, J.H., Majer E., Zuo, D., Oristaglio M. 3-D electromagnetic modeling and nonlinear inversion. Geophysics, 65(3): 804–822 (2000)
22. 
Xie, G.Q., Li, J.H., Lin, C.C. New SGILD EM modeling and inversion in Geophysics and Nano-Physics: Three Dimensional Electromagnetics, 2: 193–213 (2002)Google Scholar
23. 
Xie, G.Q., Li, J., Xie, L., Xie, F., Li, J.H. The 3D GL EM-Flow-Heat-Stress Coupled Modeling. PIERS Online, 3(4): 411–417 (2007)
24. 
Xie, G.Q., Li, J.H., Xie, L., Xie, F. A GL metro carlo EM inversion. Journal of Electromagnetic Waves and Applications, 20(14): 1991–2000 (2006)
25. 
Xie, G.Q., Li, J.H., Xie, F., Xie, Lee. 3D GL EM and quantum mechanical coupled modeling for the nanometer materials. PIERS Online, 3(4): 418–422 (2007)
26. 
Xie, G.Q., Li, J.H., Xie, L., Xie, F. GL EM modeling and inversion based on the new EM integral equation. Report of GLGEO patent, 2005, No. GLP05001, p. 38–96Google Scholar
27. 
Xie, G.Q., Li, J.H., Li, J., Xie, F. 3D and 2.5D AGLID EMS stirring modeling in the cylindrical coordinate system. PIERS Online, 2(5): 505–509 (2006)
28. 
Xie, G.Q., Li, J.H., Xie, F. A GLEMFCS coupled modeling and inversion for icing disaster on high voltage lines, Proceeding of PIERS 2008 in Hangzhou, March 24–28Google Scholar
29. 
Xie, G.Q., Li, J.H., Xie, L., Xie, F. GLEM mechanical and acoustic field time domain modeling for materials and exploration with dispersion. Proceeding of PIERS 2008, p.354–364Google Scholar
30. 
Xie, G.Q., Lin, C.C., Li, J. GILD EM modeling in nanometer material using magnetic field integral equation. J. Mathmatica Applicata, 16: 149–156 (2003)
31. 
Xie, G.Q., Xie, F., Xie, L., Li, J. New GL method and its advantages for resolving historical difficulties. Progress In Electromagnetics Research, PIER, 63: 141–152 (2006) 