This talk surveys interdisciplinary research at the University of Illinois on spacetime discontinuous Galerkin (SDG) methods implemented on adaptive grids that are full unstructured in both space and time.  SDG technology may be advantageous in problems requiring high-resolution, dynamic solutions; it offers linear computational complexity in the number of spacetime elements, element-wise balance properties with respect to spacetime Riemann fluxes, arbitrarily high-order accuracy (for smooth solutions) on compact stencils, favorable intrinsic stability properties, and an asynchronous, scalable structure for parallel computation.

This talk surveys interdisciplinary research at the University of Illinois on spacetime discontinuous Galerkin (SDG) methods implemented on adaptive grids that are full unstructured in both space and time.  SDG technology may be advantageous in problems requiring high-resolution, dynamic solutions; it offers linear computational complexity in the number of spacetime elements, element-wise balance properties with respect to spacetime Riemann fluxes, arbitrarily high-order accuracy (for smooth solutions) on compact stencils, favorable intrinsic stability properties, and an asynchronous, scalable structure for parallel computation.

The presentation begins with a review of SDG fundamentals, including a coordinate-free development of balance laws for general spacetime control volumes using differential forms and an interleaved spacetime mesh generation and finite-element solution procedure.  Applications to elastodynamic fracture and hyperbolic heat conduction demonstrate the effectiveness of the SDG methodology.  In the fracture application, we highlight the use of adaptive spacetime meshing to track solution-dependent crack growth and a new two-scale, delayed-damage cohesive model.  We close with a new sharp-interface model for concurrent coupling of continuum and atomistic models of the dynamics of solids.  Implementing the standard balance laws and compatibility relations in the SDG framework is sufficient to achieve reflection-free coupling, and the need for various algorithmic devices, such as overlap regions and various forms of non-physical damping that are commonly used to suppress spurious reflections, is eliminated.

The presentation begins with a review of SDG fundamentals, including a coordinate-free development of balance laws for general spacetime control volumes using differential forms and an interleaved spacetime mesh generation and finite-element solution procedure.  Applications to elastodynamic fracture and hyperbolic heat conduction demonstrate the effectiveness of the SDG methodology.  In the fracture application, we highlight the use of adaptive spacetime meshing to track solution-dependent crack growth and a new two-scale, delayed-damage cohesive model.  We close with a new sharp-interface model for concurrent coupling of continuum and atomistic models of the dynamics of solids.  Implementing the standard balance laws and compatibility relations in the SDG framework is sufficient to achieve reflection-free coupling, and the need for various algorithmic devices, such as overlap regions and various forms of non-physical damping that are commonly used to suppress spurious reflections, is eliminated.