A spectral mimetic least-squares method for the Stokes equations with no-slip boundary condition

Abstract

Formulation of locally conservative least-squares finite element methods (LS-FEM) for the Stokes equations with the no-slip boundary condition has been a long standing problem. Existing LSFEMs that yield exactly divergence free velocities require nonstandard boundary conditions [5], while methods that admit the no-slip condition satisfy the incompressibility equation only approximately [6, Chapter 7]. Here we address this problem by proving a new non-standard stability bound for the velocity-vorticity-pressure Stokes system augmented with a no-slip boundary condition. This bound gives rise to a norm-equivalent least-squares functional in which the velocity can be approximated by div-conforming finite element spaces, thereby enabling a locally-conservative approximations of this variable. We also provide a practical realization of the new LSFEM using high-order spectral mimetic finite element spaces [24] and report several numerical tests, which confirm its mimetic properties.

Exact spatial and temporal balance of energy exchanges within a horizontally explicit/vertically implicit non-hydrostatic atmosphere

“…As a more complex example, consider the expression ∇ψ h , w h , with w h ∈ W h . In a similar way, this expression corresponds to ∇ψ h , w h = Ω ∇ψ h w h dΩ , which is a column vector of d W elements with element j, ∇ψ h , w h j , given by [16] ∇ψ h , w h j…”

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].…”