Error analysis of a collocation method on graded meshes for a fractional Laplacian problem

被引：2

作者：

Chen, Minghua ^{[1
]}

Deng, Weihua ^{[1
]}

Min, Chao ^{[1
]}

Shi, Jiankang ^{[1
]}

Stynes, Martin ^{[2
]}

机构：

[1] Lanzhou Univ, Sch Math & Stat, Gansu Key Lab Appl Math & Complex Syst, Lanzhou 730000, Peoples R China

[2] Beijing Computat Sci Res Ctr, Appl & Computat Math Div, Beijing 100094, Peoples R China

来源：

ADVANCES IN COMPUTATIONAL MATHEMATICS | 2024年 / 50卷 / 03期

基金：

中国国家自然科学基金;

关键词：

Fractional Laplacian; Collocation method; Graded meshes; Error analysis; FINITE-DIFFERENCE METHOD; REGULARITY; DOMAINS;

D O I：

10.1007/s10444-024-10146-3

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

The numerical solution of a 1D fractional Laplacian boundary value problem is studied. Although the fractional Laplacian is one of the most important and prominent nonlocal operators, its numerical analysis is challenging, partly because the problem's solution has in general a weak singularity at the boundary of the domain. To solve the problem numerically, we use piecewise linear collocation on a mesh that is graded to handle the boundary singularity. A rigorous analysis yields a bound on the maximum nodal error which shows how the order of convergence of the method depends on the grading of the mesh; hence, one can determine the optimal mesh grading. Numerical results are presented that confirm the sharpness of the error analysis.

引用

页数：27

共 21 条

[1] REGULARITY THEORY AND HIGH ORDER NUMERICAL METHODS FOR THE (1D)-FRACTIONAL LAPLACIAN [J].