Localized Chebyshev and MLS collocation methods for solving 2D steady state nonlocal diffusion and peridynamic equations

The localized Chebyshev collocation method and MLS collocation method are presented to obtain the solution of two-dimensional nonlocal diffusion and peridynamic equations. The Chebyshev polynomial and MLS interpolation techniques are used to construct shape functions in the frame of the collocation method. Low computational cost and high accuracy are the main advantages of these two methods for solving nonlocal diffusion and peridynamic equations. Several numerical examples are provided to show the validity and applicability of the proposed method with the regular and irregular domains. Numerical experiments indicate that the localized Chebyshev collocation method has high accuracy for nonlocal problems with continuous solutions. The MLS collocation method is more efficient and can maintain good behavior for problems with discontinuous solutions.