The immersed Element-free Galerkin method for interface problems

Interface problems appear in many fields of science. They have attracted much attention from theorists and numerical analysts over the years. Computationally, many numerical methods designed for smooth solutions do not work well. In this paper, the modified Element-free Galerkin (EFG) method with the immersed interface method is presented for a two-dimensional interface problem, which models the contact between two materials with different heat conductivities. The adjustment of background grids in the EFG method is the key to this problem. The numerical scheme of this problem is given in this paper. Finally, two numerical examples are presented to verify the modified EFG method. Keywords: the immersed interface method; the Element-free Galerkin (EFG) method; background grids.

[1]  Ming-Wan Lu,et al.  Least‐squares collocation meshless method , 2001 .

[2]  Satya N. Atluri,et al.  A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method , 1998 .

[3]  C. Peskin.  Numerical analysis of blood flow in the heart , 1977 .

[4]  T. Belytschko,et al.  Fracture and crack growth by element free Galerkin methods , 1994 .

[5]  Zhilin Li,et al.  The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains (Frontiers in Applied Mathematics) , 2006 .

[6]  John B. Bell,et al.  A Cartesian Grid Projection Method for the Incompressible Euler Equations in Complex Geometries , 1994, SIAM J. Sci. Comput..

[7]  T. Belytschko,et al.  Crack propagation by element-free Galerkin methods , 1995 .

[8]  S. Osher,et al.  A level set approach for computing solutions to incompressible two-phase flow , 1994 .

[9]  Gregory E. Fasshauer,et al.  Meshfree Approximation Methods with Matlab , 2007, Interdisciplinary Mathematical Sciences.

[10]  B. Nayroles,et al.  Generalizing the finite element method: Diffuse approximation and diffuse elements , 1992 .

[11]  T. Belytschko,et al.  DYNAMIC FRACTURE USING ELEMENT-FREE GALERKIN METHODS , 1996 .

[12]  R. Temam.  Navier-Stokes Equations , 1977 .

[13]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[14]  G. R. Shubin,et al.  An analysis of the grid orientation effect in numerical simulation of miscible displacement , 1984 .