Stabilized equal low-order finite elements in ice sheet modeling – accuracy and robustness

Abstract

We investigate the accuracy and robustness of one of the most common methods used in glaciology for finite element discretization of the p-Stokes equations: linear equal order finite elements with Galerkin least-squares (GLS) stabilization on anisotropic meshes. Furthermore, we compare the results to other stabilized methods. We find that the vertical velocity component is more sensitive to the choice of GLS stabilization parameter than horizontal velocity. Additionally, the accuracy of the vertical velocity component is especially important since errors in this component can cause ice surface instabilities and propagate into future ice volume predictions. If the element cell size is set to the minimum edge length and the stabilization parameter is allowed to vary non-linearly with viscosity, the GLS stabilization parameter found in literature is a good choice on simple domains. However, near ice margins the standard parameter choice may result in significant oscillations in the vertical component of the surface velocity. For these reasons, other stabilization techniques, in particular the interior penalty method, result in better accuracy and are less sensitive to the choice of stabilization parameter. During this work, we also discovered that the manufactured solutions often used to evaluate errors in glaciology are not reliable due to high artificial surface forces at singularities. We perform our numerical experiments in both FEniCS and Elmer/Ice.

A full Stokes subgrid scheme in two dimensions for simulation of grounding line migration in ice sheets using Elmer/ICE (v8.3)

“…We employ linear Lagrange elements with Galerkin least-square (GLS) stabilization (Franca and Frey, 1992;Helanow and Ahlkrona, 2018) to avoid spurious oscillations in the pressure using the standard MISMIP setting in Elmer/ICE (Durand et al, 2009a;Gagliardini et al, 2013) approximating solutions in the spaces V k and Q k * in Eq. 13.…”

Extended enthalpy formulations in the Ice-sheet and Sea-level System Model (ISSM) version 4.17: discontinuous conductivity and anisotropic streamline upwind Petrov–Galerkin (SUPG) method

“…transport according to Darcy's law, are equally feasible (e.g. Fowler, 1984;Hewitt and Schoof, 2017). Sensible heat flux within the temperate ice is assumed to be small compared to heat production due to deformation and regarded as a source term in Eq.…”

“…Modern state-of-the-art ice sheet models (ISMs) simulate the thermal state according to the enthalpy method originally formulated in Aschwanden et al (2012) and further developed and verified in Kleiner et al (2015), Blatter and Greve (2015), Greve and Blatter (2016), and Hewitt and Schoof (2017). The main advantage of this formulation is the elimination of tracking the CTS, as both cold and temperate ice domains are handled within one equation for the enthalpy E; temperature T and liquid water fraction ω are diagnostically computed from enthalpy.…”

A full Stokes subgrid model for simulation of grounding line migration in ice sheets using Elmer/ICE(v8.3)

“…We employ linear Lagrange elements with Galerkin Least Square (GLS) stabilization Franca and Frey (1992); Helanow and Ahlkrona (2018) to avoid spurious oscillations in the pressure using the standard setting in Elmer/ICE Gagliardini et al (2013) approximating solutions in the spaces V k and M k * in Eq. (13).…”