In this thesis, we investigate the application of duality-basedgoal-oriented adaptive error control to the computation ofshear stresses in biomedical flow problems. As a model problem,we consider the linear Stokes equations.Adaptive error control for goal functionals expressed as surfaceintegrals, as in the case of shear stresses, require theformulation of a dual Stokes problem where the shear stressgoal functional enters as a driving force. This may lead toinstabilities (oscillations) in the dual pressure. A partial solutionto this problem is to reformulate surface integrals asvolume integrals.We fi nd that the volume formulation leads to signi cant improvements,both for the stability of the dual pressure and the quality of effi ciency indices. Various strategies for approximationof the dual problem, mesh re finement, and representationsof shear stress goal functionals are examined.The strategies have been implemented in Python based onthe FEniCS/DOLFIN framework and applied to a pair oftwo-dimensional geometries, including a simple test case onthe unit square with known primal and dual analytic solutions,and an idealized model of an aneurysm geometry.