Abstract
In this thesis, we investigate the application of duality-based goal-oriented adaptive error control to the computation of shear stresses in biomedical flow problems. As a model problem, we consider the linear Stokes equations.
Adaptive error control for goal functionals expressed as surface integrals, as in the case of shear stresses, require the formulation of a dual Stokes problem where the shear stress goal functional enters as a driving force. This may lead to instabilities (oscillations) in the dual pressure. A partial solution to this problem is to reformulate surface integrals as volume integrals.
We find that the volume formulation leads to significant improvements, both for the stability of the dual pressure and the quality of efficiency indices. Various strategies for approximation of the dual problem, mesh refinement, and representations of shear stress goal functionals are examined. The strategies have been implemented in Python based on the FEniCS/DOLFIN framework and applied to a pair of two-dimensional geometries, including a simple test case on the unit square with known primal and dual analytic solutions, and an idealized model of an aneurysm geometry.