Asymptotically compatible piecewise quadratic polynomial collocation for nonlocal model

10/19/2020

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by   Minghua Chen, et al.

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In this paper, we propose and analyze piecewise quadratic polynomial 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 δ=𝒪(h^β), β≥0 are provided. More concretely, the global error is 𝒪(h^max{2,4-2β}) if δ is the grid point, but it shall drop down to 𝒪(h^min{2,1+β}) if δ is not the grid point. In particular, the asymptotically compatible scheme are also rigorous proved, which has the global error 𝒪(h^min{2,2β}) as δ,h→ 0. Finally, numerical experiments are presented to verify the theoretical results.