Generalized Finite Difference Method For Elliptic Interface Problems

In this article a generalized finite difference method(GFDM),which is a meshless method based on Taylor series expansions and weighted moving least squares,is proposed to solve the elliptic interface problem and the elasticity interface problem.This method turns original elliptic interface problem to be two coupled elliptic non-interface subproblems and these subproblems can yield a large sparse matrices.The solutions are found by solving coupled elliptic subproblems with sparse coefficient matrix,which significantly improves the efficiency for the interface problem,especially for the complex geometric interface.Furthermore,based on the key idea of GFDM which can approximate the derivatives of unknown variables by linear summation of nearby nodal value,we can deal well with the interface condition which have jump derivatives by using the GFDM.Ten numerical examples are provided to illustrate the features of the proposed method,including the acceptable accuracy and the efficiency.Similarly,this method turns the original elasticity interface problem to be four coupled elasticity non-interface subproblems and a large sparse matrix can be yielded by those subproblems.This method makes the interface become a part of boundaries of subproblems,it can deal well with problems with complex geometrical interfaces.Three numerical examples are provided to verify the accuracy and stability of the proposed method when the elasticity interface problems are considered.Numerical results show the H¹ error of the GFDM has the same convergence rate as the ₂L error and the size of jumps on interface only has little effect on the stability of this method.