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

Abstract

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.