Quadrature rules for finite element approximations of 1D nonlocal problems

详细信息    查看全文

• 作者：Xiaoping Zhang^a ; ^{xpzhang.math@whu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Max Gunzburger^b ; ^{mgunzburger@fsu.edu" class="auth_mail" title="E-mail the corresponding author} ; Lili Ju^c ; ^d ; ^{ju@math.sc.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Nonlocal problems ; Finite element approximations ; Finite-part integrals ; Product rules ; DE rules
• 刊名：Journal of Computational Physics
• 出版年：2016
• 出版时间：1 April 2016
• 年：2016
• 卷：310
• 期：Complete
• 页码：213-236
• 全文大小：933 K

文摘

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.