Finite elements numerical solution of a coupled profile–velocity–temperature shallow ice sheet approximation model

Abstract

This work deals with the numerical solution of a complex mathematical model arising in theoretical glaciology. The global moving boundary problem governs thermomechanical processes jointly with ice sheet hydrodynamics. One major novelty is the inclusion of the ice velocity field computation in the framework of the shallow ice model so that it can be coupled with profile and temperature equations. Moreover, the proposed basal velocity and shear stress laws allow the integration of basal sliding effects in the global model. Both features were not taking into account in a previous paper (Math. Model. Methods Appl. Sci. 12 (2) (2002) 229) and provide more realistic convective terms and more complete Signorini boundary conditions for the thermal problem. In the proposed numerical algorithm, one- and two-dimensional piecewise linear Lagrange finite elements in space and a semi-implicit upwinding scheme in time are combined with duality and Newton's methods for nonlinearities. A simulation example involving real data issued from Antarctic shows the temperature, profile and velocity qualitative behaviour as well as the free boundaries and basal effects.