Achieving Textbook Multigrid Efficiency for Hydrostatic Ice Sheet Flow

Abstract

Abstract. The hydrostatic equations for ice sheet flow offer improved fidelity compared with the shallow ice approximation and shallow stream approximation popular in today's ice sheet models. Nevertheless, they present a serious bottleneck because they require the solution of a threedimensional (3D) nonlinear system, as opposed to the two-dimensional system present in the shallow stream approximation. This 3D system is posed on high-aspect domains with strong anisotropy and variation in coefficients, making it expensive to solve with current methods. This paper presents a Newton-Krylov multigrid solver for the hydrostatic equations that demonstrates textbook multigrid efficiency (an order of magnitude reduction in residual per iteration and solution of the fine-level system at a small multiple of the cost of a residual evaluation). Scalability on Blue Gene/P is demonstrated, and the method is compared to various algebraic methods that are in use or have been proposed as viable approaches. 1. Introduction. The dynamic response of ice streams and outlet glaciers is poorly represented by the shallowness assumptions inherent in the present generation of ice sheet models. Accurate simulation of this response is crucial for predicting sea level rise; indeed, the inability of available models, based on the shallow ice approximation (SIA) [21] and shallow stream approximation (SSA) [26,38], to simulate these processes was cited as a major deficiency in the Fourth Assessment Report of the Intergovernmental Panel on Climate Change [3].The hydrostatic equations were introduced in [4] as a model of intermediate complexity between the full non-Newtonian Stokes system and the vertically integrated SIA and SSA models. A more precise analysis, including the limiting cases of fast and slow sliding, was given in [33]. Well-posedness was proven in [8], and approximation properties of finite-element methods were analyzed in [14,7]. Hindmarsh [20] conducted a perturbation analysis in one horizontal dimension to study the validity of different continuum models including SIA, hydrostatic, and Stokes. In [30], a model intercomparison was presented using several implementations of the

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