A least-squares finite element method for a nonlinear Stokes problem in glaciology

A stationary Stokes problem with nonlinear rheology and with mixed no-slip and sliding basal boundary conditions is considered. The model describes the flow of ice in glaciers and ice sheets. A least-squares finite element method is developed and analyzed. The method does not require that the finite element spaces satisfy an inf-sup condition. Moreover, the usage of negative Sobolev norm in the least-squares functional allows for the use of standard piecewise polynomials spaces for both the velocity and pressure approximations. A Picard-type iterative method is used to linearize the Stokes problem. It is shown that the linearized least-squares functional is coercive and continuous in an appropriate solution space so the existence and uniqueness of a weak solution immediately follows as do optimal error estimates for finite element approximations. Numerical tests are provided to illustrate the theory.