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**A discontinuous \(hp\) finite element method for the Euler and Navier-Stokes equations.**
*(English)*
Zbl 0985.76048

Summary: This paper introduces a new method for the solution of Euler and Navier-Stokes equations, which is based on the application of a recently developed discontinuous Galerkin technique to obtain a compact, higher-order accurate and stable solver. The method involves a weak imposition of continuity conditions on the state variables and on inviscid and diffusive fluxes across inter-element and domain boundaries. Within each element the field variables are approximated using polynomial expansions with local support; therefore, this method is particularly amenable to adaptive refinements and polynomial enrichment. Moreover, the order of spectral approximation on each element can be adaptively controlled according to the regularity of the solution. The particular formulation on which the method is based makes possible a consistent implementation of boundary conditions, and the approximate solutions are locally (elementwise) conservative. The results of numerical experiments for representative benchmarks suggest that the method is robust, capable of delivering high rates of convergence, and well suited to be implemented in parallel computers.

### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

76N15 | Gas dynamics (general theory) |

### Keywords:

discontinuous \(hp\) finite element method; Navier-Stokes equations; Euler equations; compact stable solver; higher-order solver; discontinuous Galerkin technique; polynomial expansions
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\textit{C. E. Baumann} and \textit{J. T. Oden}, Int. J. Numer. Methods Fluids 31, No. 1, 79--95 (1999; Zbl 0985.76048)

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