Incompressible non-Newtonian fluids with shear-thinning behavior are of great interest in many applications including ice modeling. For example the Glen model uses a power law relation between the stress tensor and the symmetric velocity gradient. Monotone constitutive relations of this sort and also non-explicit relations that describe yielding fit into the general framework of implicit constitutive laws. Within this general framework convergence of finite element approximations can be obtained by compactness arguments. We highlight the difficulties in the numerical analysis and present some results.

 

In this context the p-Laplace equation serves as a useful toy model for the development of tools in the numerical analysis. Adaptive finite element methods are of particular relevance for solutions with low regularity and in situations, where geometric features have to be resolved. For meshes generated by adaptive refinement we show recent results on their grading and the implications for time-dependent problems.

 

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