Surface elevation errors in finite element Stokes models for glacier evolution

TL;DR

The paper addresses error analysis for a Stokes-based glacier evolution model formulated as a nonlinear variational inequality with a regularized surface-motion term. It develops an abstract FE error bound for nonlinear operator VIs on Banach spaces and applies it to a glacier problem, yielding a concrete bound on surface-elevation error in terms of bed discretization, velocity solves, and surface representation. The analysis hinges on conjectures about Lipschitz dependence of the surface velocity on the surface and a 4-coercivity property of the regularized surface-motion operator, with numerical experiments providing partial support. This work provides a rigorous framework for understanding discretization errors in fully coupled, non-shallow glacier models and informs mesh design and bed-topography treatment for improved accuracy.

Abstract

The primary data which determine the evolution of glaciation are the bedrock elevation and the surface mass balance. From this data, which we assume is defined over a fixed land region, the glacier's geometry solves a free boundary problem which balances the time derivative of the surface elevation, the surface velocity from the Stokes flow of the ice, and the surface balance rate. This problem can be posed in weak form as a variational inequality over a cone of admissible surface elevation functions, those which are above the bedrock topography. After some preparatory theory for the Stokes problem, we conjecture that the corresponding continuous-space, implicit time-step variational inequality problem is well-posed if the surface kinematical equation is appropriately regularized. This conjecture is supported by physical arguments and numerical evidence. We then prove a general theorem which bounds the numerical error made by finite element approximations of nonlinear-operator variational inequalities in Banach spaces. This bound is a sum of error terms of different types, special to variational inequalities. When it is applied to the implicit time-step glacier problem there are three terms in the bound: an error from discretizing the bed elevation, an error from numerically solving for the Stokes velocity, and finally an expected error which is quasi-optimal in the finite element space representation of the surface elevation. The design of glacier models is then reconsidered based on this a priori error analysis.

Figures (10)

• Figure 1: Glacier notation: $x\in\Omega\subset\mathbb{R}^2$ and $(x,z) \in \Lambda(t)\subset\mathbb{R}^3$.
• Figure 1: Glacier margins with finite-slope "wedge" shapes (left) are possible, while shallow theory has fractional powers (center). Actual glacier margins might have overhangs and fractures (right).
• Figure 1: Left: If $b_h=\pi_h b$ is the interpolant of $b$ then generally $\mathcal{K}_h\not\subset\mathcal{K}$. Right: Defining $b_h=R_h^{\oplus} b$, using \ref{['eq:monotoneop']}, implies $b_h\ge b$ and $\mathcal{K}_h\subset\mathcal{K}$.
• Figure 1: The Stokes problem for the velocity $\mathbf{u}_h$ is solved on a meshed polygonal domain, between $P_1$ functions $b_h$ and $s_h$. Computation of $F_{\Delta t}^{\epsilon,h}$ requires evaluation of the surface trace $\mathbf{u}_h|_{s_h}$.
• Figure 1: For a smooth bedrock elevation $b$ with local minima, one surface elevation state $s_1$ fills-in the minima with flat ice (left), while the other is ice-free $s_2=b$ (right). Because neither state generates flow, $\mathbf{u}|_{s_i}=\bm{0}$, the surface motion map $\Phi$ in equation \ref{['eq:definePhi']} cannot be coercive.

Theorems & Definitions (27)

• Lemma 2.1
• Lemma 2.2: Trace inequality
• Lemma 2.3: Surface velocity bound
• Lemma 3.1: Bound on surface motion
• Proof 1
• Conjecture 3.2: Surface velocity is Lipschitz in the surface elevation
• Definition 3.3
• Lemma 3.4
• Proof 2
• Definition 3.5