This presentation begins with a review of the asynchronous Spacetime Discontinuous Galerkin (aSDG) method. In its original context, the aSDG method exploits the characteristic structure of hyperbolic PDEs and special asynchronous spacetime meshes to generate solution schemes with linear computational complexity in the number of spacetime finite elements. In lieu of conventional time marching, the solution advances through local implicit solutions on patches (small clusters of spacetime elements) to combine the stability of implicit schemes with the linear complexity and locality of explicit methods. Remeshing operations can be similarly localized to obtain adaptive solutions that are exceptionally responsive to solution dynamics. In addition, aSDG solution schemes are rich in embarrassingly parallel structure that is ripe for exploitation on HPC platforms. Mathematical underpinnings of the method, such as the use of exterior calculus and differential forms in objective spacetime formulations, as well as an advanced application to dynamic fracture complete the review.
The latter part of the presentation describes ongoing research on a new parallel–adaptive aSDG scheme. We set aside established techniques and abstractions, such as the domain decomposition method (DDM) and the bulk synchronous parallel (BSP) model of parallel computation. The result is a novel parallel finite element method that is free of synchronization barriers (major obstacles to effecient use of exascale platforms) and that vastly simplifies load-balancing in response to dynamic adaptive meshing. Other current research directions, including prospects for applying aSDG methods to elliptic and parabolic systems and extensions of spacetime meshing to three spatial dimensions will be covered in brief.