Mimetic Spectral Element Advection

Abstract

We present a discretization of the linear advection of differential forms on bounded domains. The framework established in [4] is extended to incorporate the Lie derivative, L, by means of Cartan's homotopy formula. The method is based on a physics-compatible discretization with spectral accuracy . It will be shown that the derived scheme has spectral convergence with local mass conservation. Artificial dispersion depends on the order of time integration.

Mimetic Spectral Element Method for Anisotropic Diffusion

“…Possible extensions to spectral methods were described by Robidoux, [109]. A different approach for constructing arbitrary order mimetic finite elements has been proposed by the authors [31,64,92,96], with applications to advection problems [95], Stokes' flow [81], MHD equilibrium [94], Navier-Stokes [93], and within a Least-Squares finite element formulation [16,62,63,91].…”

A Geometric Approach Towards Momentum Conservation

“…Now that we understand how momentum density should be integrated over a volume, we can also define convection of momentum density. After pairing with an arbitrary vector field, w, we obtain a volume form and we apply the Lie derivative to this volume form, see [11]. The Lie derivative for a volume form, β (n) , is given by…”

“…1, and P m w is the matrix which maps discrete velocity (which is discretized as mass-fluxes) to discrete momentum (on the staggered-grid). The discrete representation of momentum-flux, pressure force i w p (n) and traction forces, µ (∇ w v) (which can be equivalently writ- • Convective-flux, see [11], F…”

Mixed Mimetic Spectral Element Method Applied to Darcy’s Problem

“…In this paper we will make use of the spectral element method described in [9,19], application of these ideas to Stokes' flow see [16][17][18]; Poisson equation for volume forms [22]; advection equation [21]; derivation of a momentum conservation scheme [24]. Extension to compatible isogeometric methods see [11,12].…”