My primary research line is related to numerical global atmospheric modelling, mainly using unstructured polygonal spherical grids (such as the one shown below) to solve the differential equations.
In real application, we could have locally refined grids, such as the one shown in the next figure.
I have been particularly interested in how certain grid structures interfere in the solution of the PDEs. The figures below shows the error of a usual finite volume discretization of the divergence operator for o solid body rotation flow. Note that there are well defined grid related structures, called grid imprinting. See Peixoto and Barros (2013) for details here.
I have also worked with a semi-Lagrangian transport model in this kind of grid. The figure below shows a results of a deformational test case example. See Peixoto and Barros (2014) for details here.
More recently, I have been working with a modification of mimetic finite volume schemes for the shallow water equations, to allow better accuracy (see Peixoto (2016) for details here). Below are results for the barotropically unstable jet test case.
Please see more details of the things I have been investigating in my papers with collaborators. Do not hesitate to contact me if you wish to discuss anything related to this research topic.
You can find the software I developed to work on these grids and all the experiments up until the shallow water model here.