Analysis of (shifted) piecewise quadratic polynomial collocation for nonlocal diffusion model

Abstract

The piecewise quadratic polynomial collocation is used to approximate the nonlocal model, which generally leads to a nonsymmetric indefinite system (Chen et al. (2021) [5]). In this case, the discrete maximum principle is not satisfied, which might be trickier for the stability analysis of the high-order numerical schemes (D'Elia et al. (2020) [10]; Leng et al. (2021) [26]). Here, we present a modified (shifted-symmetric) piecewise quadratic polynomial collocation for solving the linear nonlocal diffusion model, which leads to a symmetric positive definite system and satisfies the discrete maximum principle. Using Faulhaber's formula and Riemann zeta function, the perturbation error for symmetric positive definite system and nonsymmetric indefinite system are given. Then rigorous convergence analysis for the nonlocal models are provided under the general horizon parameter δ=O(hβ), with β≥0. More concretely, the global error is O(hmin⁡{2,1+β}) if δ is not set as a grid point, while it recovers O(hmax⁡{2,4−2β}) when δ is set as a grid point. We also prove that the shifted-symmetric scheme is asymptotically compatible, which has the global error O(hmin⁡{2,2β}) as δ,h→0. The numerical experiments (including two-dimensional case) are performed to verify the convergence.