Anshu Kumar | Searcharxiv

arXiv2026

Optimal Control in Hilbert Complex Spaces with Finite Element Exterior Calculus

We develop a framework for PDE-constrained optimal control on Hilbert complexes and its structure-preserving discretization by finite element exterior calculus (FEEC), with emphasis on problems where nontrivial topology creates physically meaningful global modes. We separate the gauge-fixed local state from harmonic components representing global circulation or flux, and introduce finite-dimensional topological actuators whose reachable modes are determined jointly by domain topology and actuator rank. We establish well-posedness, weak optimality conditions, reachability results, and FEEC error estimates preserving the harmonic structure. Numerical experiments on contractible and multiply connected domains demonstrate circulation and cavity- flux control, rank-dependent reachability, mesh-independent optimization, and topology-induced singular convergence. We further extend the framework to time-dependent Maxwell control, where harmonic electric circulation and magnetic flux become dynamical states exhibiting explicit conservation and reachability properties.

math.OC↗

arXiv2025

Survey of Mathematical Models of and Numerical Methods for Fluid Dynamics Water Engineering

Computational fluid dynamics (CFD) has become a cornerstone of modern water engineering, providing quantitative tools for the analysis, prediction, and management of complex hydraulic systems across a wide range of spatial and temporal scales. This survey reviews the mathematical models and numerical methods that underpin CFD applications in water engineering, from depth-averaged formulations such as the shallow water equations to fully three-dimensional Navier-Stokes models, as well as selected alternative modeling approaches. We examine the historical development of these models, their mathematical structure, and the numerical discretization and solution strategies commonly employed in practice, including finite difference, finite volume, and finite element methods. Beyond core solver technology, the survey addresses practical modeling issues such as source-term treatment, wetting and drying, turbulence modeling, free-surface representation, and computational efficiency. The growing role of data integration is also discussed, encompassing data assimilation, uncertainty quantification, and emerging machine-learning-assisted approaches that complement physics-based solvers. To illustrate the impact of modeling and numerical choices on real-world applications, representative case studies from large-scale water management systems are reviewed. By integrating theory, numerical techniques, and applied perspectives, this survey provides a unified reference for researchers and practitioners seeking to understand both the foundational principles and contemporary challenges of CFD in water engineering.

physics.flu-dyn↗