UC San Diego

This dissertation studies the analysis and numerical approximation of nonlocal operators with generalized interaction neighborhoods. Its primary focus is on half-space interactions, with additional results for heterogeneous localization and cone interactions. Half-space operators arise in peridynamic correspondence models and are useful for nonlocal formulations of diffusion and first-order systems. Based on the theoretical foundation for half-space operators, the convergence rate estimates are obtained as nonlocal horizon and discretization parameter tend to zero. The analysis is then generalized to noncoercive problems.Chapter 3 lays a theoretical foundation for half-space operators. It establishes distributional definitions of the nonlocal gradient and divergence, identifies the associated nonlocal Sobolev spaces through Fourier symbols, and proves density, Poincar´e, compactness, and uniform Poincar´e estimates. These results provide the stability tools as nonlocality vanishes.Chapters 4 and 5 study convergence rates at continuous and discrete levels. At the continuous level, the nonlocal solutions are shown to converge to the corresponding local solution, and direct residual estimates give convergence rates under the stated regularity and consistency hypotheses. The discrete theory gives qualitative asymptotic compatibility under explicit density and stability assumptions and combines modeling and Galerkin errors to obtain finite element rate estimates for the half-space problem. Numerical experiments illustrate the predicted behavior and identify places where the theoretical rates are not yet sharp.Chapter 6 extends the discussion beyond coercive scalar problems. It gives conditional mixed and double adaptivity frameworks and proves continuous and discrete convergence rates for a T-coercive modified Poisson formulation. Chapter 7 develops heterogeneous localization and proves a Besov trace theorem on a strip that supports the formulation of nonlocal transmission problems. Chapter 8 extends the half-space construction to cones whose asymmetric part lies in a hemisphere, proving a cone symbol estimate, a Poincar´e inequality, and vanishing-horizon compactness for a fixed scaled cone under the stated kernel assumptions.