Saving FLOPs in Geophysics with optimal p-adaptivity

Summary A Discontinuous Galerkin Spectral Element (DGSE) method is presented for highly accurate seismic modelling with an emphasis on Floating Point Operations (FLOPs) saving. While standard finite-element-based applications feature unstructured meshes, the scheme developed in this work relies on regular Cartesian meshes. The spatial mesh refinement (h-adaptivity) is avoided and instead, an ad-hoc p-adaptivity is proposed. This alleviates the task of mesh building and avoids unnecessary FLOPs related to mesh over-refinements. A protocol to determine the appropriate polynomial order for each element depending on the local wave velocity is defined. This protocol allows to maintain the error of the numerical solution below an arbitrary threshold. In a model representative of sand environments, with extreme low velocity in the shallow region, the p-adaptive DGSE yields an effective saving of FLOPs by an order of magnitude compared to non-adaptive DGSE and by a factor of 4.4 compared to the Spectral Element Method (SEM) for 2D elastic modelling in sand environments. This makes p-adaptive DGSE an attractive approach to efficiently tackle large velocity contrasts.