Goal-Oriented Mesh Adaptation for Firedrake

Mesh adaptation can be a very powerful tool for improving the accuracy and/or efficiency of simulations, as has been documented extensively in the literature. However, it is still not widely used. This is largely because commonly available approaches require experience and an in-depth understanding of both the problem at hand and its interaction with the discretisation method being used. Many users of numerical software lack in one of these, and sometimes both. The goal-oriented mesh adaptation framework reduces the required level of experience, since it is driven by error estimates determined by the PDE and a user-specified quantity of interest (QoI) that we seek to accurately approximate, such as the power output of a turbine or the drag on an aeroplane wing. This massively simplifies the challenge of using mesh adaptation in an optimal manner, since it is now only required to choose a QoI, which will generally be somewhat obvious and straightforward to implement.

The primary output of this eCSE is a new goal-oriented error estimation and mesh adaptation module, Pyroteus, which handles the solution of PDEs and their adjoints on sequences of meshes and enables the automatic computation of error indicators associated with such equations. In addition, both the PETSc solver library and Firedrake finite element library have been extended to support metric-based mesh adaptation functionality that can be used by Pyroteus to generate anisotropic adapted meshes based on the goal-oriented error indicators it computes.