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This paper presents a fully discrete numerical scheme for one-dimensional nonlocal wave equations and provides a rigorous theoretical analysis. To facilitate the spatial discretization, we introduce an auxiliary variable analogous to the gradient field in local discontinuous Galerkin (DG) methods for classical partial differential equations (PDEs) and reformulate the equation into a system of equations. The proposed scheme then uses a DG method for spatial discretization and the Crank-Nicolson method for time integration. We prove optimal L2 error convergence for both the solution and the auxiliary variable under a special class of radial kernels at the semi-discrete level. In addition, for general kernels, we demonstrate the asymptotic compatibility of the scheme, ensuring that it recovers the classical DG approximation of the local wave equation in the zero-horizon limit. Furthermore, we prove that the fully discrete scheme preserves the energy of the nonlocal wave equation. A series of numerical experiments are presented to validate the theoretical findings. 

This paper investigates solution strategies for nonlinear problems in Hilbert spaces, such as nonlinear partial differential equations (PDEs) in Sobolev spaces, when only finite measurements are available. We formulate this as a nonlinear optimal recovery problem, establishing its well-posedness and proving its convergence to the true solution as the number of measurements increases. However, the resulting formulation might not have a finite-dimensional solution in general. We thus present a sufficient condition for the finite dimensionality of the solution, applicable to problems with well-defined point evaluation measurements. To address the broader setting, we introduce a relaxed nonlinear optimal recovery and provide a detailed convergence analysis. An illustrative example is given to demonstrate that our formulations and theoretical findings offer a comprehensive framework for solving nonlinear problems in infinite-dimensional spaces with limited data. 

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.