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### Abstract

We address the free boundary problem that consists in finding the shape of a three dimensional glacier over a given period and under given climatic conditions. Glacier surface moves by sliding, internal shear and external exchange of mass. Ice is modelled as a non Newtonian fluid. Given the shape of the glacier, the velocity of ice is obtained by solving a stationary non-linear Stokes problem with a sliding law along the bedrock-ice interface. The shape of the glacier is updated by computing a Volume Of Fluid (VOF) function, which satisfies a transport equation. Climatic effects (accumulation and ablation of ice) are taken into account in the source term of this equation. A decoupling algorithm with a two-grid method allows the velocity of ice and the VOF to be computed using different numerical techniques, such that a Finite Element Method (FEM) and a characteristics method. On a theoretical level, we prove the well-posedness of the non-linear Stokes problem. A priori estimates for the convergence of the FEM are established by using a quasi-norm technique. Eventually, convergence of the linearisation schemes, such that a fixed point method and a Newton method, is proved. Several applications demonstrate the potential of the numerical method to simulate the motion of a glacier during a long period. The first one consists in the simulation of Rhone et Aletsch glacier from 1880 to 2100 by using climatic data provided by glaciologists. The glacier reconstructions over the last 120 years are validated against measurements. Afterwards, several different climatic scenarios are investigated in order to predict the shape the glaciers until 2100. A dramatic retreat during the 21st century is anticipated for both glaciers. The second application is an inverse problem. It aims to find a climate parametrization allowing a glacier to fit some of its moraines. Two other aspects of glaciology are also addressed in this thesis. The first one consists in modeling and in simulating ice collapse during the calving process. The previous ice flow model is supplemented by a Damage variable which describes the presence of micro crack in ice. An additional numerical scheme allows the Damage field to be solved and a two dimensional simulation of calving to be performed. The second problem aims to prove the existence of stationary ice sheet when considering shallow ice model and a simplified geometry. Numerical investigation confirms the theoretical result and shows physical properties of the solution.