Abstract
The aim of this paper is to study a nonlinear stationary Stokes problem with mixed boundary conditions that describes the ice velocity and pressure fields of grounded glaciers under Glen's flow law. Using convex analysis arguments, we prove the existence and the uniqueness of a weak solution. A finite element method is applied with approximation spaces that satisfy the inf-sup condition, and a priori error estimates are established by using a quasinorm technique. Several algorithms (including Newton's method) are proposed to solve the nonlinearity of the Stokes problem and are proved to be convergent. Our results are supported by numerical convergence studies.
Highlights
In this paper we consider a model problem that is commonly used by glaciologists to compute the motion of glaciers
All of them have considered a simplified model, called first-order approximation 2. This model is obtained by rewriting the Stokes equations into a dimensionless form and by dropping all terms of order O 2, where is the typical aspect ratio of glaciers. This simplification results into a nonlinear elliptic problem for the horizontal velocity field, the vertical component, and the pressure field being determined a posteriori
Colinge and Rappaz first demonstrated the well-posedness of this problem and proved the convergence of Advances in Numerical Analysis the finite element approximation with piecewise linear continuous functions in 3
Summary
In this paper we consider a model problem that is commonly used by glaciologists to compute the motion of glaciers. Glen’s law and the mass momentum equation lead to a nonlinear stationary Stokes problem with a strain-dependent viscosity. All of them have considered a simplified model, called first-order approximation 2. This model is obtained by rewriting the Stokes equations into a dimensionless form and by dropping all terms of order O 2 , where is the typical aspect ratio of glaciers. This simplification results into a nonlinear elliptic problem for the horizontal velocity field, the vertical component, and the pressure field being determined a posteriori. Inspired by the work of Baranger/Najib 4 and Barrett/Liu 5 on non-Newtonian problems, a priori and a posteriori error estimates were obtained later in 6–8
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