Adjoint methods have become extremely useful tools for computational geoscience. In this thesis, the two main implementations – the continuous and discrete adjoint methods – are compared within the same code framework, Firedrake, in terms of accuracy, computational cost and usability. The discrete adjoint method is deemed best suited for use within gradient-based optimisation routines and is used to invert timeseries data for a tsunami source, with different automatic differentiation tools applied to the source and propagation models.

The second focus is on mesh adaptation techniques suitable for representing time-dependent, multi-scale coastal ocean dynamics. Historically, h- and r-adaptation communities have developed methods independently. However, the metric-based framework is independent of how the discretisation is modified, meaning that it allows for h-, r- and hybrid adaptation. It has the advantage of allowing control of anisotropy, which is demonstrated to be useful for advection-dominated problems. A monitor-based approach – specific to r-adaptation and built upon optimal transport theory – is also considered and is shown to be capable of reducing point-wise error in a sediment transport test case, at comparable overall cost to uniform meshing.

Much time and effort has been spent on mesh adaptation research in the past four decades.
However, it is arguably under-utilised in practical applications. This is largely because making effective choices of error estimator, optimal mesh model and adaptation method generally requires both experience and an in-depth understanding of the problem at hand. The two strands of research, on adaptation and adjoint methods, come together most clearly in goal-oriented mesh adaptation, which goes some way to reduce user experience requirements. Meshes are constructed based on accurately approximating a diagnostic quantity of interest, with the adjoint method used to evaluate associated error estimators. The culmination of this thesis applies goal-oriented mesh adaptation to four geoscience problems of increasing complexity.