Numerical analysis of space-time hybridized discontinuous Galerkin methods for incompressible flows

Many industrial problems require the solution of the incompressible Navier-Stokes equations on moving and deforming domains. Notable examples include the simulation of rotating wind turbines in strong air flow, wave impact on offshore structures, and arterial blood flow in the human body. A viable candidate for the numerical solution of the Navier-Stokes equations on time-dependent domains is the space-time discontinuous Galerkin (DG) method, which makes no distinction between spatial and temporal variables. Space-time DG is well suited to handle moving and deforming domains but at a significant increase in computational cost in comparison to traditional time-stepping methods. Attempts to rectify this situation have led to the pairing of space-time DG with the hybridized discontinuous Galerkin (HDG) method, which was developed to reduce the computational expense of DG. The combination of the two methods results in a scheme that retains the high-order spatial and temporal accuracy and geometric flexibility of space-time DG at a reduced cost. Moreover, the use of hybridization allows for the design of pressure-robust space-time methods on time-dependent domains, which is a class of mimetic methods that inherit at the discrete level a fundamental invariance property of the incompressible Navier-Stokes equations. The space-time HDG method has been successfully applied to incompressible flow problems on time-dependent domains; however, at present, no supporting theoretical analysis can be found in the literature. This thesis is a first step toward such an analysis. In particular, we perform a thorough theoretical convergence analysis of a space-time HDG method for the incompressible Navier-Stokes equations on fixed domains, and of a space-time HDG method for the linear advection-diffusion equation on time-dependent domains. The former contribution elucidates the difficulties involved in the theoretical analysis of space-time HDG methods for the Navier-Stokes equations, while the latter contribution introduces a framework for the convergence analysis of space-time HDG methods on time-dependent domains. We begin with an a priori error analysis of a pressure-robust HDG method for the stationary Navier-Stokes equations. Then, we provide an a priori error analysis of a pressure-robust space-time HDG method from which we conclude that the space-time HDG method converges to strong solutions of the Navier-Stokes equations. This leaves open the question of convergence to weak solutions, which we answer in the affirmative using compactness techniques. Finally, we provide an a priori error analysis of a space-time HDG method for the linear advection-diffusion equation on time-dependent domains

Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).Peer ReviewedPostprint (author's final draft

A Comparison of Hybridized and Standard DG Methods for Target-Based hp-Adaptive Simulation of Compressible Flow

We present a comparison between hybridized and non-hybridized discontinuous Galerkin methods in the context of target-based hp-adaptation for compressible flow problems. The aim is to provide a critical assessment of the computational efficiency of hybridized DG methods. Hybridization of finite element discretizations has the main advantage, that the resulting set of algebraic equations has globally coupled degrees of freedom only on the skeleton of the computational mesh. Consequently, solving for these degrees of freedom involves the solution of a potentially much smaller system. This not only reduces storage requirements, but also allows for a faster solution with iterative solvers. Using a discrete-adjoint approach, sensitivities with respect to output functionals are computed to drive the adaptation. From the error distribution given by the adjoint-based error estimator, h- or p-refinement is chosen based on the smoothness of the solution which can be quantified by properly-chosen smoothness indicators. Numerical results are shown for subsonic, transonic, and supersonic flow around the NACA0012 airfoil. hp-adaptation proves to be superior to pure h-adaptation if discontinuous or singular flow features are involved. In all cases, a higher polynomial degree turns out to be beneficial. We show that for polynomial degree of approximation p=2 and higher, and for a broad range of test cases, HDG performs better than DG in terms of runtime and memory requirements

eXtended hybridizable discontinuous Galerkin for incompressible flow problems with unfitted meshes and interfaces

The eXtended hybridizable discontinuous Galerkin (X-HDG) method is developed for the solution of Stokes problems with void or material interfaces. X-HDG is a novel method that combines the hybridizable discontinuous Galerkin (HDG) method with an eXtended finite element strategy, resulting in a high-order, unfitted, superconvergent method, with an explicit definition of the interface geometry by means of a level-set function. For elements not cut by the interface, the standard HDG formulation is applied, whereas a modified weak form for the local problem is proposed for cut elements. Heaviside enrichment is considered on cut faces and in cut elements in the case of bimaterial problems. Two-dimensional numerical examples demonstrate that the applicability, accuracy, and superconvergence properties of HDG are inherited in X-HDG, with the freedom of computational meshes that do not fit the interfacesPeer ReviewedPostprint (author's final draft

An embedded--hybridized discontinuous Galerkin finite element method for the Stokes equations

We present and analyze a new embedded--hybridized discontinuous Galerkin finite element method for the Stokes problem. The method has the attractive properties of full hybridized methods, namely an $H({\rm div})$-conforming velocity field, pointwise satisfaction of the continuity equation and \emph{a priori} error estimates for the velocity that are independent of the pressure. The embedded--hybridized formulation has advantages over a full hybridized formulation in that it has fewer global degrees-of-freedom for a given mesh and the algebraic structure of the resulting linear system is better suited to fast iterative solvers. The analysis results are supported by a range of numerical examples that demonstrate rates of convergence, and which show computational efficiency gains over a full hybridized formulation

An embedded-hybridized discontinuous Galerkin method for the coupled Stokes-Darcy system

We introduce an embedded-hybridized discontinuous Galerkin (EDG-HDG) method for the coupled Stokes-Darcy system. This EDG-HDG method is a pointwise mass-conserving discretization resulting in a divergence-conforming velocity field on the whole domain. In the proposed scheme, coupling between the Stokes and Darcy domains is achieved naturally through the EDG-HDG facet variables. \emph{A priori} error analysis shows optimal convergence rates, and that the velocity error does not depend on the pressure. The error analysis is verified through numerical examples on unstructured grids for different orders of polynomial approximation