Stokes equations. The space discretization of the inviscid terms of the Navier-Stokes equations is constructed fol-This paper deals with a high-order accurate discontinuous finite element method for the numerical solution of the compressible lowing the ideas described in the works of Cockburn et Navier-Stokes equations. We extend a discontinuous finite element al., except for the limiting procedure. As shown, for examdiscretization originally considered for hyperbolic systems such as ple, in [6], no limiting is in fact needed even for inviscid the Euler equations to the case of the Navier-Stokes equations by flows, provided that the solution is sufficiently smooth. The treating the viscous terms with a mixed formulation. The method space discretization of the viscous terms is constructed by combines two key ideas which are at the basis of the finite volume and of the finite element method, the physics of wave propagation resorting to a mixed finite element formulation. Howbeing accounted for by means of Riemann problems and accuracy ever, instead of using the Raviart-Thomas formulation as being obtained by means of high-order polynomial approximations in [13, 14], we approximate both the unknown and its within elements. As a consequence the method is ideally suited to gradient in the same discontinuous function space (see compute high-order accurate solution of the Navier-Stokes equa-Section 2).
tions on unstructured grids. The performance of the proposed method is illustrated by computing the compressible viscous flow
The method combines different features commonly assoon a flat plate and around a NACA0012 airfoil for several flow reciated to finite element and to finite volume methods. As gimes using constant, linear, quadratic, and cubic elements. ᮊ 1997 in classical finite element methods, in fact, accuracy is ob-
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tained by means of high-order polynomial approximation within an element rather than by wide stencils as in the case of finite volume schemes. The physics of wave propa-267