h, r, and hr adaptivity with applications in numerical ocean modelling

The purpose of this article is to introduce techniques for performing h, r, and hr adaptivity in the context of numerical ocean modelling. These supplements to a standard numerical discretization offer the possibility of focusing degrees of freedom, and hence computational resources, in certain important regions. Full flexibility of the definition of these regions may be supplied through a priori and a posteriori error measures, or through consideration of the physics of the underlying problem. One of the most significant advantages present with the use of unstructured meshes in ocean modelling is the possibilities this opens up for the use of adaptive algorithms. Indeed it is hard to envisage unstructured mesh based models becoming a serious viable alternative to models based upon structured meshes until the efficient and robust use of adaptive meshing is employed. Examples of the use of adaptive methods are presented here for free surface flows and circulation in the Mediterranean Sea. Practical issues relating to mesh quality, including anisotropy, element shape, discretization errors, and matrix conditioning are also discussed.