Quadrature rules for finite element approximations of 1D nonlocal problems

JOURNAL OF COMPUTATIONAL PHYSICS

卷:310

Article

Zhang, Xiaoping^1,2  Gunzburger, Max³  Ju, Lili^4,5 

[1] Wuhan Univ, Sch Math & Stat, Wuhan 430072, Hubei, Peoples R China

[2] Wuhan Univ, Computat Sci Hubei Key Lab, Wuhan 430072, Hubei, Peoples R China

[3] Florida State Univ, Dept Sci Comp, Tallahassee, FL 32306 USA

[4] Univ S Carolina, Dept Math, Columbia, SC 29208 USA

[5] Beijing Computat Sci Res Ctr, Beijing 100193, Peoples R China

关键词: Nonlocal problems;    Finite element approximations;    Finite-part integrals;    Product rules;    DE rules;   

DOI  :  10.1016/j.jcp.2016.01.016

来源: Elsevier

【 摘 要 】

It is well known that calculations of the entries of the stiffness matrix in the finite element approximations of nonlocal diffusion and mechanics models are often very time-consuming due to the double integration process over the domain and the singularities of the nonlocal kernel functions. In this paper, we propose some effective and accurate quadrature rules for computing these double integrals for one-dimensional nonlocal problems; in particular, for problems with highly singular kernels, the corresponding inner integrals can be first evaluated exactly in our method, and the outer one then will be approximated by some popular quadrature rules. With these quadrature rules, the assembly of the stiffness matrix in the finite element method for the nonlocal problems becomes similar to that for the classical partial differential equations and is thus quite efficient. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free