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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. 

Abstract We prove two compactness results for function spaces with finite Dirichlet energy of half‐space nonlocal gradients. In each of these results, we provide sufficient conditions on a sequence of kernel functions that guarantee the asymptotic compact embedding of the associated nonlocal function spaces into the class of square‐integrable functions. Moreover, we will demonstrate that the sequence of nonlocal function spaces converges in an appropriate sense to a limiting function space. As an application, we prove uniform Poincaré‐type inequalities for sequence of half‐space gradient operators. We also apply the compactness result to demonstrate the convergence of appropriately parameterized nonlocal heterogeneous anisotropic diffusion problems. We will construct asymptotically compatible schemes for these type of problems. Another application concerns the convergence and robust discretization of a nonlocal optimal control problem. 

Abstract Oceanic transform faults play an essential role in plate tectonics. Yet to date, there is no unifying explanation for the global trend in broad-scale transform fault topography, ranging from deep valleys to shallow topographic highs. Using three-dimensional numerical models, we find that spreading-rate dependent magmatism within the transform domain exerts a first-order control on the observed spectrum of transform fault depths. Low-rate magmatism results in deep transform valleys caused by transform-parallel tectonic stretching; intermediate-rate magmatism fully accommodates far-field stretching, but strike-slip motion induces across-transform tension, producing transform strength dependent shallow valleys; high-rate magmatism produces elevated transform zones due to local compression. Our models also address the observation that fracture zones are consistently shallower than their adjacent transform fault zones. These results suggest that plate motion change is not a necessary condition for reproducing oceanic transform topography and that oceanic transform faults are not simple conservative strike-slip plate boundaries. 

In this paper, we consider a class of discontinuous Galerkin (DG) methods for one-dimensional nonlocal diffusion (ND) problems. The nonlocal models, which are integral equations, are widely used in describing many physical phenomena with long-range interactions. The ND problem is the nonlocal analog of the classic diffusion problem, and as the interaction radius (horizon) vanishes, then the nonlocality disappears and the ND problem converges to the classic diffusion problem. Under certain conditions, the exact solution to the ND problem may exhibit discontinuities, setting it apart from the classic diffusion problem. Since the DG method shows its great advantages in resolving problems with discontinuities in computational fluid dynamics over the past several decades, it is natural to adopt the DG method to compute the ND problems. Based on [Q. Du, L. Ju, J. Lu and X. Tian,Commun. Appl. Math. Comput. 2 (2020) 31–55], we develop the DG methods with different penalty terms, ensuring that the proposed DG methods have local counterparts as the horizon vanishes. This indicates the proposed methods will converge to the existing DG schemes as the horizon vanishes, which is crucial for achievingasymptotic compatibility. Rigorous proofs are provided to demonstrate the stability, error estimates, and asymptotic compatibility of the proposed DG schemes. To observe the effect of the nonlocal diffusion, we also consider the time-dependent convection–diffusion problems with nonlocal diffusion. We conduct several numerical experiments, including accuracy tests and Burgers’ equation with nonlocal diffusion, and various horizons are taken to show the good performance of the proposed algorithm and validate the theoretical findings. 

This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with general kernel functions (integrable or fractional type with finite or infinite supports) and study the associated nonlocal vector identities. We study the nonlocal function space on bounded domains associated with zero Dirichlet boundary conditions and the half-ball gradient operator and show it is a separable Hilbert space with smooth functions dense in it. A major result is the nonlocal Poincaré inequality, based on which a few applications are discussed, and these include applications to nonlocal convection–diffusion, nonlocal correspondence model of linear elasticity and nonlocal Helmholtz decomposition on bounded domains. 

Abstract Oceanic detachment faults represent an end-member form of seafloor creation, associated with relatively weak magmatism at slow-spreading mid-ocean ridges. We use 3-D numerical models to investigate the underlying mechanisms for why detachment faults predominantly form on the transform side (inside corner) of a ridge-transform intersection as opposed to the fracture zone side (outside corner). One hypothesis for this behavior is that the slipping, and hence weaker, transform fault allows for the detachment fault to form on the inside corner, and a stronger fracture zone prevents the detachment fault from forming on the outside corner. However, the results of our numerical models, which simulate different frictional strengths in the transform and fracture zone, do not support the first hypothesis. Instead, the model results, combined with evidence from rock physics experiments, suggest that shear-stress on transform fault generates excess lithospheric tension that promotes detachment faulting on the inside corner. 

Motivated by some variational problems from a nonlocal model of mechanics, this work presents a set of sufficient conditions that guarantee a compact inclusion in the function space of $$ L^{p} $$ vector fields defined on a domain $$ \Omega $$ that is either a bounded domain in $$ \mathbb{R}^{d} $$ or $$ \mathbb{R}^{d} $$ itself. The criteria are nonlocal and are given with respect to nonlocal interaction kernels that may not be necessarily radially symmetric. Moreover, these criteria for vector fields are also different from those given for scalar fields in that the conditions are based on nonlocal interactions involving only parts of the components of the vector fields. The $$ L^{p} $$ compactness criteria are utilized in demonstrating the convergence of minimizers of parameterized nonlocal energy functionals. 

This work aims to prove a Hardy-type inequality and a trace theorem for a class of function spaces on smooth domains with a nonlocal character. Functions in these spaces are allowed to be as rough as an [Formula: see text]-function inside the domain of definition but as smooth as a [Formula: see text]-function near the boundary. This feature is captured by a norm that is characterized by a nonlocal interaction kernel defined heterogeneously with a special localization feature on the boundary. Thus, the trace theorem we obtain here can be viewed as an improvement and refinement of the classical trace theorem for fractional Sobolev spaces [Formula: see text]. Similarly, the Hardy-type inequalities we establish for functions that vanish on the boundary show that functions in this generalized space have the same decay rate to the boundary as functions in the smaller space [Formula: see text]. The results we prove extend existing results shown in the Hilbert space setting with p = 2. A Poincaré-type inequality we establish for the function space under consideration together with the new trace theorem allows formulating and proving well-posedness of a nonlinear nonlocal variational problem with conventional local boundary condition.