## $$p$$-multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations.(English)Zbl 1177.76194

Summary: We present a $$p$$-multigrid solution algorithm for a high-order discontinuous Galerkin finite element discretization of the compressible Navier-Stokes equations. The algorithm employs an element line Jacobi smoother in which lines of elements are formed using coupling based on a $$p = 0$$ discretization of the scalar convection-diffusion equation. Fourier analysis of the two-level $$p$$-multigrid algorithm for convection-diffusion shows that element line Jacobi presents a significant improvement over element Jacobi especially for high Reynolds number flows and stretched grids. Results from inviscid and viscous test cases demonstrate optimal $$h^{p + 1}$$ order of accuracy as well as $$p$$-independent multigrid convergence rates, at least up to $$p = 3$$. In addition, for the smooth problems considered, $$p$$-refinement outperforms $$h$$-refinement in terms of the time required to reach a desired high accuracy level.

### MSC:

 76M10 Finite element methods applied to problems in fluid mechanics 76D05 Navier-Stokes equations for incompressible viscous fluids

Wesseling
Full Text:

### References:

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