AIR for a space-time hybridizable discontinuous Galerkin method

Abdullah Ali Sivas (University of Waterloo)
Ben Southworth (University of Colorado, Boulder)
Sander Rhebergen (University of Waterloo)

Space-time finite element methods are excellent for the discretization of partial differential equations (PDEs), including on time-dependent domains. Unlike classical time-stepping methods, such as Runge-Kutta or multistep methods, space-time methods make no distinction between spatial and temporal variables. Instead, the PDE is discretized directly in d+1-dimensional space-time, where d is the spatial dimension. Consider, for example, the time-dependent advection equation in d spatial dimensions,

    \[\partial_t u + {\bf a}\cdot\nabla u = f.\]

To apply the space-time finite element method, we introduce first the space-time gradient


and space-time advective velocity

    \[\tilde{{\bf a}}=(1,{\bf a}).\]

We, then, write the time-dependent advection equation as a `steady’ advection equation in space-time:

    \[\tilde{{\bf a}}\cdot\tilde{\nabla}u = f.\]

We discretize this equation by the space-time HDG method of [1,3].

In this talk, we discuss the solution of the space-time HDG discretization of the advection and advection-diffusion equation on time-dependent domains by lAIR algebraic multigrid [2]. lAIR was shown in [2] to be an optimal solver for hyperbolic and advection-dominated problems. This makes lAIR ideal also as a solver for space-time discretizations of advection dominated flows. We will also discuss and compare the solution of space-time HDG discretizations resulting from an all-at-once discretization, in which the d+1-dimensional space-time domain has been discretized into a d+1-dimensional unstructured mesh, and a slab approach, in which the space-time domain is first partitioned into time-slabs and the problem is solved one slab at a time. We investigate the efficiency of lAIR for purely hyperbolic and strongly advection-dominated problems, which are difficult or intractable for many parallel-in-time methods, and also consider the weakly advection-dominated case. We furthermore investigate lAIR in combination with space-time adaptive mesh refinement, a unique advantage of space-time finite elements over a traditional separation of space and time.

[1] K.L.A. Kirk, et al., Analysis of a space-time hybridizable discontinuous Galerkin method for the advection-diffusion problem on time-dependent domains, SIAM J. Numer. Anal., 57/4 (2019).

[2] T. A. Manteuffel, et al., Nonsymmetric algebraic multigrid based on local approximate ideal restriction AIR, SIAM J. Sci. Comput., 40/6 (2018).

[3] S. Rhebergen and B. Cockburn, Space-time hybridizable discontinuous Galerkin method for the advection-diffusion equation on moving and deforming meshes, in The Courant-Friedrichs-Lewy (CFL) condition, 80 years after its discovery, Birkhauser Science, 2013.