ASYMPTOTICALLY COMPATIBLE SCHEMES AND APPLICATIONS TO ROBUST DISCRETIZATION OF NONLOCAL MODELS

作者：Tian, Xiaochuan^*; Du, Qiang

来源：SIAM Journal on Numerical Analysis, 2014, 52(4): 1641-1665.

DOI：10.1137/130942644

• nonlocal diffusion
• peridynamic models
• state-based model
• local limit
• finite element
• convergence analysis
• asymptotically compatible scheme

Many problems in nature, being characterized by a parameter, are of interest both with a fixed parameter value and with the parameter approaching an asymptotic limit. Numerical schemes that are convergent in both regimes offer robust discretizations, which can be highly desirable in practice. The asymptotically compatible schemes studied in this paper meet such objectives for a class of parametrized problems. An abstract mathematical framework is established rigorously here together with applications to the numerical solution of both nonlocal models and their local limits. In particular, the framework can be applied to nonlocal diffusion models and a general state-based peridynamic system parametrized by the horizon radius. Recent findings have exposed the risks associated with some discretizations of nonlocal models when the horizon radius is proportional to the discretization parameter. Thus, it is desirable to develop asymptotically compatible schemes for such models so as to offer robust numerical discretizations to problems involving nonlocal interactions on multiple scales. This work provides new insight in this regard through a careful analysis of related conforming finite element discretizations and the finding is valid under minimal regularity assumptions on exact solutions. It reveals that as long as the finite element space contains continuous piecewise linear functions, the Galerkin finite element approximation is always asymptotically compatible. For piecewise constant finite element, whenever applicable, it is shown that a correct local limit solution can also be obtained as long as the discretization (mesh) parameter decreases faster than the modeling (horizon) parameter does. These results can be used to guide future computational studies of nonlocal problems.

• 出版日期2014