High-Resolution Multimesh hp-FEM for Simulation of Compressible Particle-Laden Gas Flows

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The objective of this project is the development of a new approach to hp-adaptivity for scalar transport equations and hyperbolic systems on the basis of methods implemented during the first funding period. The proposed variational multiscale framework will combine a continuous Galerkin approximation of coarse scales with a discontinuous approximation of fine scales. To this end, continuous (P1 or Q1) finite elements will be enriched by Taylor basis functions that have been employed to design hierarchichal slope limiting techniques for p-adaptive discontinuous Galerkin (DG) methods. At the coarse scale level, the discrete maximum principle will be enforced using an algebraic flux correction scheme of FCT type (Flux-Corrected Transport). The multimesh hp-FEM approach implemented in the software package HERMES makes it possible to use individually designed meshes for each variable and each level of the p-hierarchy. The type of adaptivity (h or p) is determined using maximum principles to estimate the local regularity. The process of smoothness and error estimation involves the use of reconstruction techniques for the highest-order derivatives. The decomposition of the nonlinear system into a global problem for the coarse scales and small local problems for the fine scales offers substantial efficiency gains. The extension of the new hp-FEM to the Euler equations and equations of the two-fluid modell will build on the previously developed numerical algorithms for coarse-scale simulations.