arXiv subjects
Publications and source records attributed to Jai Tushar.
We design and analyze virtual element methods for a quad-curl problem on general polygonal domains that are based on the Hodge decomposition of divergence-free vector fields. Numerical results that corroborate the theoretical analysis are also presented.
Hybrid finite element methods such as hybridizable discontinuous Galerkin, hybrid high-order and weak Galerkin have emerged as powerful techniques for solving partial differential equations on general polytopal meshes. Despite their diverse mathematical origins, these methods share a common computational structure involving hybrid discrete spaces, local projection operators and static condensation. This work presents a comprehensive framework for implementing such methods within the Gridap finite element library. We introduce new abstractions for polytopal mesh representation using graph-based structures, broken polynomial spaces on arbitrary mesh entities, patch-based local assembly for cell-wise linear systems, high-level local operator construction and automated static condensation. These abstractions enable concise implementations of hybrid methods while maintaining computational efficiency through Julia's just-in-time compilation and Gridap's lazy evaluation strategies. We demonstrate the framework through implementations of several non-conforming polytopal methods for the Poisson problem, linear elasticity, incompressible Stokes flow and optimal control on polytopal meshes.
In this work, we build on the discrete trace theory developed by Badia, Droniou, and Tushar (Foundations of Computational Mathematics, in press, 2025; \href{https://doi.org/10.1007/s10208-025-09734-6}{doi:10.1007/s10208-025-09734-6}) to analyze the convergence rate of the Balancing Domain Decomposition by Constraints (BDDC) preconditioner generated from non-conforming polytopal hybrid discretizations. We prove polylogarithmic bounds on the condition number for the preconditioner that are independent of the mesh parameter and the number of subdomains, and that hold on polytopal meshes. The analysis relies on the continuity of a face truncation operator, which we establish in the fully discrete polytopal setting. To validate the theory, we present numerical experiments that confirm the truncation estimate and condition number bounds. In particular, we conduct weak scalability tests for second-order elliptic problems discretized using discontinuous skeletal methods, specifically Hybridizable Discontinuous Galerkin (HDG) and Hybrid High-Order (HHO) methods. We also demonstrate the robustness of the preconditioner for piecewise discontinuous coefficients with large jumps.
In this work, we study the control constrained distributed optimal control of a stationary doubly diffusive flow model. For the control problem, we use a well-posedness analysis based on minimal assumptions on data and domain. We show the existence of an optimal control with quadratic type cost functional, study the Fréchet differentiability properties of the control-to-state map and establish the first-order necessary optimality conditions corresponding to the optimal control problem. Expanding on this we prove the local optimality of a reference control using second-order sufficient optimality condition for the control problem.
In this work we develop a discrete trace theory that spans non-conforming hybrid discretization methods and holds on polytopal meshes. A notion of a discrete trace seminorm is defined, and trace and lifting results with respect to a discrete $H^1$-seminorm on the hybrid fully discrete space are proven. Building on these results we also prove a truncation estimate for piecewise polynomials in the discrete trace seminorm. Finally, we conduct two numerical tests in which we compute the proposed discrete operators and investigate their spectrum to verify the theoretical analysis. The development of this theory is motivated by the design and analysis of preconditioners for hybrid methods, e.g., of substructuring domain decomposition type.
We study a stationary model of doubly diffusive flows with temperature-dependent viscosity on bounded Lipschitz domains in two and three dimensions. A new well-posedness and regularity analysis of weak solutions under minimal assumptions on domain geometry and data regularity are established. A fully non-conforming finite element method based on Crouzeix-Raviart elements, which ensures locally exactly divergence-free velocity fields is explored. Unlike previously proposed schemes, this discretization enables to establish uniqueness of the discrete solutions. We prove the well-posedness of the discrete problem and derive a priori error estimates. An accuracy test is conducted to verify the theoretical error decay rates in flow, Stokes and Darcy regimes on convex and non-convex domains, and a benchmark test of flow in a porous cavity is conducted, comparing the proposed method with existing literature.