Analysis Of Shifted-symmetric Collocation Method For Nonlocal Model

As is well known,piecewise quadratic polynomial collocation is used to approxi-mate the nonlocal model,which derives the nonsymmetric indefinite system.Here,we present the modified(shifted-symmetric)collocation method for nonlocal model,which has the symmetric positive definite system.In this work,we propose and analyze piece-wise quadratic polynomial shifted-symmetric collocation for solving the linear nonlocal diffusion model with the weakly singular kernels.The detailed proof of the convergence analysis for the nonlocal models with the horizon parameter ?=O(h?),??0 are provided.More concretely,the global error is O(hmax{2,4-2?})if ? is the grid point,but it shall drop down to O(hmin{2,1+?})if ? is not the grid point.In particular,the asymptotically compatible scheme are also rigorous proved,which has the global error O(hmin{2,2?})as ?,h?0.Finally,numerical experiments are presented to verify the theoretical results.