Anisotropic Goal-Oriented Mesh Adaptation in Firedrake

We consider metric-based mesh adaptation methods for steady-state partial differential equations (PDEs), solved using the finite element method in Firedrake. In this work, a number of mesh-adaptive methods are implemented within this framework, each enabling accurate approximation of a scalar quantity of interest (QoI). Through the QoI we define adjoint equations, with which we may gain understanding of its sensitivities to aspects of the PDE solution. Dual weighted residual type error estimation techniques are utilised in order to enable a goal-oriented strategy. Isotropic and anisotropic approaches are considered, both of which are able to achieve the same relative error in approximating the QoI as with uniform refinement, but using fewer elements. For validation purposes, we compare QoI values resulting from these approaches against analytical values which may be extracted for a particular advection-diffusion based test case. Potential applications in desalination plant outfall modelling are discussed.