In this article we present the first results on domain decomposition methods for nonlocal operators. We present a nonlocal variational formulation for these operators and establish the well-posedness of associated boundary value problems, proving a nonlocal Poincaré inequality. To determine the conditioning of the discretized operator, we prove a spectral equivalence which leads to a mesh size independent upper bound for the condition number of the stiffness matrix. We then introduce a nonlocal two-domain variational formulation utilizing nonlocal transmission conditions, and prove equivalence with the single-domain formulation. A nonlocal Schur complement is introduced. We establish condition number bounds for the nonlocal stiffness and Schur complement matrices. Supporting numerical experiments demonstrating the conditioning of the nonlocal one-and two-domain problems are presented. Dirichlet boundary value problem( 1.2) Let n and d denote the dimensions of the function space and the spatial domain, respectively. Ω ⊂ R d is a bounded domain, BΩ is given in (2.1), b is given, and u(x) ∈ R n is prescribed for x ∈ R d \Ω. We prescribe the value of u(x) outside Ω and not just on the boundary of Ω, owing to the nonlocal nature of the problem.Nonlocal models are useful where classical (local) models cease to be predictive. Examples include porous media flow [18,19,20], turbulence [21], fracture of solids, stress fields at dislocation cores and cracks tips, singularities present at the point of application of concentrated loads (forces, couples, heat, etc.), failure in the prediction of short wavelength behavior of elastic waves, microscale heat transfer, and fluid flow in microscale channels [22]. These are also cases where microscale fields are nonsmooth. Consequently, nonlocal models are also useful for multiscale modeling. Recent examples of nonlocal multiscale modeling include the upscaling of molecular dynamics to nonlocal continuum mechanics [23], and development of a rigorous multiscale method for the analysis of fiber-reinforced composites capable of resolving dynamics at structural length scales as well as the length scales of the reinforcing fibers [24]. Progress towards a nonlocal calculus is reported in [25]. Development and analysis of a nonlocal diffusion equation is reported in [26,27,28]. Theoretical developments for general class of integro-differential equation related to the fractional Laplacian are presented in [29,30,31]. Mathematical and numerical analysis for linear nonlocal peridynamic boundary problems appears in [32,33]. We discuss in §2 some specific contexts where the nonlocal operator L appears, and the assumptions placed upon L by those interpretations.

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Abstract. Parameterization of unstructured surface meshes is of fundamental importance in many applications of Digital Geometry Processing. Such parameterization approaches give rise to large and exceedingly ill-conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner.

Abstract. In this article, we examine the Bramble-Pasciak-Xu (BPX) preconditioner in the setting of local 2D and 3D mesh refinement. While the available optimality results for the BPX preconditioner have been constructed primarily in the setting of uniformly refined meshes, a notable exception is the 2D result due to Dahmen and Kunoth, which established BPX optimality on meshes produced by a restricted class of local 2D red-green refinement. The purpose of this article is to extend the original 2D Dahmen-Kunoth result to several additional types of local 2D and 3D red-green (conforming) and red (non-conforming) refinement procedures. The extensions are accomplished through a 3D extension of the 2D framework in the original Dahmen-Kunoth work, by which the question of optimality is reduced to establishing that locally enriched finite element subspaces allow for the construction of a scaled basis which is formally Riesz stable. This construction in turn rests entirely on establishing a number of geometrical properties between neighboring simplices produced by the local refinement algorithms. These properties are then used to build Riesz-stable scaled bases for use in the BPX optimality framework. Since the theoretical framework supports arbitrary spatial dimension d ≥ 1, we indicate clearly which geometrical properties, established here for several 2D and 3D local refinement procedures, must be re-established to show BPX optimality for spatial dimension d ≥ 4. Finally, we also present a simple alternative optimality proof of the BPX preconditioner on quasiuniform meshes in two and three spatial dimensions, through the use of K-functionals and H sstability of L 2 -projection for s ≥ 1. The proof techniques we use are quite general; in particular, the results require no smoothness assumptions on the PDE coefficients beyond those required for well-posedness in H 1 , and the refinement procedures may produce nonconforming meshes.Key words. finite element approximation theory, multilevel preconditioning, BPX, two and three dimensions, local mesh refinement, red and red-green refinement.AMS subject classifications. 65M55, 65N55, 65N22, 65F101. Introduction. In this article, we analyze the impact of local adaptive mesh refinement on the stability of multilevel finite element spaces and on the optimality (linear space and time complexity) of multilevel preconditioners. Adaptive refinement techniques have become a crucial tool for many applications, and access to optimal or near-optimal multilevel preconditioners for locally refined mesh situations is of primary concern to computational scientists. The preconditioners which can be expected to have somewhat favorable space and time complexity in such local refinement scenarios are the hierarchical basis (HB) method, the Bramble-PasciakXu (BPX) preconditioner, and the wavelet modified (or stabilized) hierarchical basis (WHB) method. While there are optimality results for both the BPX and WHB preconditioners in the literature, these are primarily for quasiuniform meshes and/o...

Recent applications of machine learning, in particular deep learning, motivate the need to address the generalizability of the statistical inference approaches in physical sciences. In this Letter, we introduce a modular physics guided machine learning framework to improve the accuracy of such data-driven predictive engines. The chief idea in our approach is to augment the knowledge of the simplified theories with the underlying learning process. To emphasize their physical importance, our architecture consists of adding certain features at intermediate layers rather than in the input layer. To demonstrate our approach, we select a canonical airfoil aerodynamic problem with the enhancement of the potential flow theory. We include the features obtained by a panel method that can be computed efficiently for an unseen configuration in our training procedure. By addressing the generalizability concerns, our results suggest that the proposed feature enhancement approach can be effectively used in many scientific machine learning applications, especially for the systems where we can use a theoretical, empirical, or simplified model to guide the learning module.