Asymptotically compatible schemes and applications to robust discretization of nonlocal models (Q2927826)

The authors are concerned with robust discretizations of nonlocal models for multiscale problems. Such nonlocal models can be parameterized by the horizon which measures the range of nonlocal interactions. In this context, the authors formulate a rigorous mathematical framework in order to analyze a class of asymptotically compatible schemes. The analysis is valid with natural assumptions on the solution regularity and approximation spaces. A typical example considered is a Dirichlet nonlocal constrained problem attached to a scalar nonlocal diffusion equation. For such a problem the authors show that each and every finite element discretization which contains piecewise linear functions provides an asymptotically compatible scheme. Consequently, this scheme is a robust discretization for both the nonlocal problem and its local limit. The subtle relationship between the horizon parameter and the discretization one is thoroughly analyzed. The study opens up new perspectives in numerical analysis of the nonlocal problems.