Preconditioning Hybrid Discontinuous Galerkin Schemes for Incompressible Flow and Linear Elasticity Problems

This dissertation presents a study on the development, analysis, and application of advanced numerical methods for solving partial differential equations (PDEs), focusing primarily on the Hybrid Discontinuous Galerkin (HDG) schemes. The research introduces innovative HDG schemes, including a monolithic divergence-conforming HDG (H(div)-HDG) scheme for linear fluid-structure interaction (FSI) problems, and proposes robust block-diagonal preconditioners and geometric multigrid algorithms to address the computational challenges associated with these methods. The work begins by reviewing H(div)-HDG schemes for Stokes and linear elasticity equations, subsequently proposing a novel monolithic scheme for FSI problems that ensures exactly divergence-free fluid velocity approximations and maintains energy stability. An optimal a priori error analysis and convergence rates are validated through numerical experiments. A significant part of the dissertation is devoted to overcoming the computational inefficiency in solving the condensed symmetric and indefinite global linear systems arising from these schemes. A uniform block-diagonal preconditioner is developed, demonstrating robustness with respect to mesh size and model parameters. Further, the dissertation extends to the construction of optimal geometric multigrid algorithms for the lowest order HDG schemes with numerical integration, emphasizing the equivalence between the global linear system and modified Crouzeix-Raviart discretizations. This foundational work facilitates the development of an hp-multigrid preconditioner for higher-order H(div)-HDG schemes applied to generalized Stokes and Navier-Stokes equations, showcasing robustness across mesh sizes and augmented Lagrangian parameters, and insensitivity to polynomial order variations. The research findings underscore the efficiency and adaptability of HDG methods in handling complex PDEs, contributing to the field of computational mathematics and numerical analysis. The dissertation concludes with discussions on future research directions, including the exploration of efficient parallel implementations and the extension of robust preconditioning techniques to more complex equations.