Spacetime Meshing with Adaptive Coarsening and Refinement

We propose a new algorithm for constructing finite-element meshes suitable for spacetime discontinuous Galerkin solutions of linear hyperbolic PDEs. Our new method is a generalization of the ‘Tent Pitcher’ algorithms of Üngör and Sheffer [3] and Erickson et al. [2]. Given a simplicially-meshed domain Ω in IR and a target time value T , our method constructs a mesh of the spacetime domain Ω× [0, T ] in IR using an advancing front method. Elements are added to the evolving mesh in small patches by moving a vertex of the front forward in time. Spacetime discontinuous Galerkin methods [4] allow the numerical solution within each patch to be computed as soon as the patch is created. This work introduces new mechanisms for adaptively coarsening and refining the front in response to error estimates returned by the numerical code. A change in the front induces a corresponding refinement or coarsening of future elements in the spacetime mesh.