Adaptive spacetime meshing for discontinuous Galerkin methods
Computational Geometry: Theory and Applications
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Important applications in science and engineering, such as modeling traffic flow, seismic waves, electromagnetics, and the simulation of mechanical stresses in materials, require the high-fidelity numerical solution of hyperbolic partial differential equations (PDEs) in space and time variables. Spacetime discontinuous Galerkin (SDG) finite element methods are used to solve such PDEs arising from wave propagation phenomena. To support an accurate and efficient solution procedure using SDG methods and to exploit the flexibility of these methods, we give a meshing algorithm to construct an unstructured simplicial spacetime mesh over an arbitrary simplicial space domain. Our algorithm is the first adaptive spacetime meshing algorithm suitable for efficient solution of nonlinear phenomena using spacetime discontinuous Galerkin finite element methods. Given a triangulated d-dimensional Euclidean space domain M (a simplicial complex) corresponding to time t = 0 and initial conditions of the underlying hyperbolic spacetime PDE, we construct an unstructured simplicial mesh of the ( d + 1)-dimensional spacetime domain Ω. Our algorithm uses a near-optimal number of spacetime elements, each with bounded temporal aspect ratio for any finite prefix of Ω. When d ≤ 2, our algorithm varies the size of spacetime elements to an a posteriori numerical estimate. Certain facets of our mesh satisfy gradient constraints that allow interleaving mesh generation with the SDG salver. Our meshing algorithm thus supports an efficient parallelizable solution strategy by SDG methods.