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Adjoint-based $$h$$-$$p$$ adaptive discontinuous Galerkin methods for the 2D compressible Euler equations. (English) Zbl 1391.76367
Summary: We investigate and present an adaptive discontinuous Galerkin algorithm driven by an adjoint-based error estimation technique for the inviscid compressible Euler equations. This approach requires the numerical approximations for the flow (i.e. primal) problem and the adjoint (i.e. dual) problem which corresponds to a particular simulation objective output of interest. The convergence of these two problems is accelerated by an $$hp$$-multigrid solver which makes use of an element Gauss-Seidel smoother on each level of the multigrid sequence. The error estimation of the output functional results in a spatial error distribution, which is used to drive an adaptive refinement strategy, which may include local mesh subdivision ($$h$$-refinement), local modification of discretization orders ($$p$$-enrichment) and the combination of both approaches known as $$hp$$-refinement. The selection between $$h$$- and $$p$$-refinement in the $$hp$$-adaptation approach is made based on a smoothness indicator applied to the most recently available flow solution values. Numerical results for the inviscid compressible flow over an idealized four-element airfoil geometry demonstrate that both pure $$h$$-refinement and pure $$p$$-enrichment algorithms achieve equivalent error reductions at each adaptation cycle compared to a uniform refinement approach, but requiring fewer degrees of freedom. The proposed $$hp$$-adaptive refinement strategy is capable of obtaining exponential error convergence in terms of degrees of freedom, and results in significant savings in computational cost. A high-speed flow test case is used to demonstrate the ability of the $$hp$$-refinement approach for capturing strong shocks or discontinuities while improving functional accuracy.

##### MSC:
 76M10 Finite element methods applied to problems in fluid mechanics 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 76N15 Gas dynamics (general theory)
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