A meshless numerical method based on the local boundary integral equation (LBIE) to solve linear and non-linear boundary value problems

Abstract

Meshless methods for solving boundary value problems have been extensively popularized in recent literature owing to their flexibility in engineering applications, especially for problems with discontinuities, and because of the high accuracy of the computed results. A meshless method for solving linear and non-linear boundary value problems, based on the local boundary integral equation method and the moving least squares (MLS) approximation, is discussed in the present article. In the present article, the implementation of the LBIE formulation for linear and non-linear problems with the linear part of the differential operator being the Helmholtz type, is developed. For non-linear problems, the total formulation and a rate formulation are developed for the implementation of the presently proposed method. The present method is a true meshless one, as it does not need domain and boundary elements to deal with the volume and boundary integrals, for linear as well as non-linear problems. The “companion solution” is employed to simplify the present formulation and reduce the computational cost. It is shown that the satisfaction of the essential as well as natural boundary conditions is quite simple, and algorithmically very efficient in the present LBIE approach, even when the non-interpolative MLS approximation is used. Numerical examples are presented for several linear and non-linear problems, for which exact solutions are available. The present method converges fast to the final solution with reasonably accurate results for both the unknown variable and its derivatives in solving non-linear problems. No post processing procedure is required to compute the derivatives of the unknown variable [as in the conventional boundary element method and field/boundary element method, as the solution from the present method, using the MLS approximation, is already smooth enough. The numerical results in these examples show that high rates of convergence with mesh refinement for the Sobolev norms ‖·‖ 0 and ‖·‖ 1 are achievable, and that the values of the unknown variable and its derivatives are quite accurate.