# Metric-Based hp-Adaptation Using a Continuous Mesh Model

## Outline

Computational Fluid dynamics as an analytical tool has been popular for the past few decades. In particular, higher order consistent methods like Discontinuous Galerkin(DG) are being developed at a fast pace. These schemes offer high accuracy at a cost similar to that of traditional finite volume or finite element methods.

These schemes involve solving a PDE on a given mesh. A classic trade-off is that between improving the accuracy of the solution by increasing the mesh resolution and increased computational cost. In most problems one is not interested in the flow properties of the entire domain but only in selected regions. Alternatively one may be interested in computing certain solution-dependent functionals very accurately. In either of these cases, a uniform refinement of the mesh results in wasteful use of computational resources. This is where adaptive schemes are important. They are favoured as they carefully distribute the degrees of freedom whereby improving the accuracy of the solution while also reducing the computational cost. This is true in particular when coupled with high order schemes, whose efficiency depends critically on this careful distribution of computational resources.

The current work involves developing anisotropic mesh adaptation for convection-diffusion problems. Previous studies also have shown that anisotropic meshing offers greater gain as compared to isotropic meshing. The freedom to adapt both the size and orientation of the mesh elements is highly suited for problems that exhibit strong anisotropic features. In this context, metric based adaptations have shown great promise. In metric based adaptation, one encodes the desirable mesh in a tensor field defined on the entire computational domain. A triangulation generated by conforming to this metric is one whose sides are nearly equilateral in a Riemannian sense to the specified metric.

The optimisation of the metric itself is performed using the so-called continuous interpolation error. This error estimate is define pointwise throughout the domain. The significance of having such an estimate is that it allows for the use of variational calculus to optimise the continuous mesh metric. While the task of finding the optimal discrete mesh is intractable by analytic means, finding a continuous mesh that is optimal with respect to this continuous interpolation error can be tackled using relatively simple variational calculus. Once computed, the optimal metric is passed onto one of the many metric-based mesh generators. The problem is solved again on the new optimised mesh and the iteration continues.