On the finite element approximation of a semicoercive stokes variational inequality arising in glaciology

• Published in: SIAM Journal on Numerical Analysis
• Main Authors: de Diego, G, Farrell, P, Hewitt, I
• Format: Article in Journal/Newspaper
• Language: English
• Published: Society for Industrial and Applied Mathematics 2022
• Subjects: Korn; Lagrange; Ice Sheet
• Online Access: https://doi.org/10.1137/21M1437640; https://ora.ox.ac.uk/objects/uuid:c607888d-f850-4ca2-99f9-19607bff6435

Stokes variational inequalities arise in the formulation of glaciological problems involving contact. We consider the problem of a two-dimensional marine ice sheet with a grounding line, although the analysis presented here is extendable to other contact problems in glaciology, such as that of subglacial cavitation. The analysis of this problem and its discretisation is complicated by the nonlinear rheology commonly used for modelling ice, the enforcement of a friction boundary condition given by a power law, and the presence of rigid modes in the velocity space, which render the variational inequality semicoercive. In this work, we consider a mixed formulation of this variational inequality involving a Lagrange multiplier and provide an analysis of its finite element approximation. Error estimates in the presence of rigid modes are obtained by means of a specially-built projection operator onto the subspace of rigid modes and a Korn-type inequality. These proofs rely on the fact that the subspace of rigid modes is at most one-dimensional, a property which is a consequence of the two-dimensionality of the domain. Numerical results are reported to validate the error estimates.