We present a study on asymptotically compatible Galerkin discretizations for a class of parametrized nonlinear variational problems. The abstract analytical framework is based on variational convergence, or Gamma-convergence. We demonstrate the broad applicability of the theoretical framework by developing asymptotically compatible finite element discretizations of some representative nonlinear nonlocal variational problems on a bounded domain. These include nonlocal nonlinear problems with classically-defined, local boundary constraints through heterogeneous localization at the boundary, as well as nonlocal problems posed on parameter-dependent domains.

翻译：本文研究了一类参数化非线性变分问题的渐近相容Galerkin离散方法。该抽象分析框架基于变分收敛（即Gamma收敛）。通过在有限区域上发展若干代表性非线性非局部变分问题的渐近相容有限元离散格式，我们论证了该理论框架的广泛适用性。这些模型包括：通过边界异质局部化实现经典局部边界约束的非线性非局部问题，以及定义在参数依赖域上的非局部问题。