Keegan Kirk | Applied Math, University of Waterloo
Analysis of Space-time Hybridizable Discontinuous Galerkin Methods for Incompressible Flows on Moving Domains
Many industrial problems require the solution of time-dependent partial differential 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. In contrast to problems posed on fixed domains, numerical methods for moving domains must satisfy the Geometric Conservation Law (GCL), a non-trivial feat.
A viable candidate that automatically satisfies the GCL 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 hybridizable discontinuous Galerkin (HDG) method, which was developed solely 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 without the associated computational burden.
This talk will comprise of three parts. We first discuss the error analysis of a space-time HDG method for the advection-diffusion problem on moving domains. Next, we will introduce pressure-robustness and prove pressure-robust error estimates for an HDG method for the steady-state incompressible Navier-Stokes equations. Lastly, we discuss a pressure-robust space-time HDG method for the time-dependent incompressible Navier-Stokes equations on moving domains and touch on some challenges in its analysis.