H(div) conforming and DG methods for incompressible Euler’s equations

A conservative finite element method for the incompressible Euler equations with variable density

“…In the absence of upwinding, the convergence rates for s = 1 were lower (closer to first order than to second order). These observations are consistent with those of [6,8], where the case of constant density is considered. Figure 1 plots the squared density errors |1−F (t)/F (0)|, F (t) = Ω ρ h (t) 2 dx, for the simulations in Tables 1-2 that used h = 1 4 .…”

“…The trilinear forms a h and b h are well-known: a h has been used in discretizations the incompressible Euler equations with constant density [6,8], and b h is a standard discontinous Galerkin discretization of the scalar advection operator [2]. These trilinear forms possess several properties that play an essential role in the conservative nature of our numerical method.…”

“…Moreover, the incompressibility constraint is only satisfied in a weak sense, and their construction relies on an evolution equation for √ ρu -a quantity that lacks physical meaning. Another method that is closely related to ours is the H(div)-conforming finite element spatial discretization of the incompressible Euler equations with constant density studied in [6,8]. When ρ ≡ 1, our spatial discretization reduces to the one used there, and our temporal discretization reduces to the midpoint rule used in [8].…”

Well-posedness and H(div)-conforming finite element approximation of a linearised model for inviscid incompressible flow

“…The use of H(div) conforming finite element methods for the approximation of incompressible flow at high Reynolds number has been receiving increasing attention from the research community recently [8,13,18]. By construction such methods can satisfy the divergence free condition exactly.…”

Análisis del método local discontinuo Galerkin para la ecuación de Fokker-Planck

“…En particular, nos referimos a [14], donde se presentan métodos de elemento finito multi-scala para la solución de (1) en dimensiones superiores. Dicho trabajo fue parte de la motivación del presente escrito, donde la diferencia principal corresponde a que proponemos un método de elemento finito mixto discontinuo (tal y como se hace recientemente en [13] para las ecuaciones incompresibles de Euler), el cual, dentro de sus principales ventajas, busca obtener una buena aproximación del flujo de probabilidad ("probability current"), además de la función de densidad de probabilidad. Con respecto al flujo de probabilidad, el cual describe en cada tiempo el cambio de la probabilidad en Ω (ver [16]), se resalta que el mismo también puede interpretarse, desde un punto de vista de mecánica cuán-tica, como la descripción del movimiento de una partícula en Ω (ver por ejemplo [12]).…”