Keegan Kirk | Applied Math, University of Waterloo
Analysis of a Space-time Hybridizable Discontinuous Galerkin Method for Advection-Diffusion Problems on Evolving Domains
Many important applications of fluid mechanics require the solution of time-dependent partial differential equations on evolving 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 evolving domains must satisfy the Geometric Conservation Law (GCL), a non-trivial feat.
A viable candidate proven to automatically satisfy the GCL is the space-time discontinuous Galerkin (DG) method, which makes no distinction between spatial and temporal variables. The problem is fully discretized in space and time instead of the typical method of lines treatment of time-dependent problems on fixed domains. This leads to an accurate scheme 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.
In this talk, we perform an a priori analysis of a space-time HDG method for the non-stationary advection-diffusion problem posed on a time-dependent domain. We discuss the derivation of anisotropic trace and inverse inequalities valid for moving meshes, which are essential tools for our analysis. Stability of the scheme is proven through the satisfaction of an inf-sup condition. Finally, we discuss the error analysis of the method and derive theoretical rates of convergence.