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.