Coordinate-free numerics: all your variation points for free?

In the object oriented literature the notion of a variation point is important. A variation point is where we expect to, or, in retrospect, are forced to, accommodate alternative solutions. Numerical software, such as solvers for partial differential equations (PDEs), are full of variation points. We observe this from how we speak about PDEs. Firstly, we speak about a multitude of PDEs to describe the various physical applications, e.g., the Navier-Stokes equations for fluid flow problems, the elastic wave equation for seismic simulation etc. Moreover, it is quite typical that mathematical modelling connected to a physical application may require various extensions of the ”basic” PDE used to describe the problem, in order to include more physically realistic effects. Examples are the introduction of physical diffusion in inviscid compressible fluid flow, the introduction of temperature effects in isentropic fluid flow or the consideration of porous media effects starting from the usual elastic wave motion. So the first variation point is the choice of PDE. The numerical discretisation process also obviously constitutes a rich source of variation points. As an example, the common numerical methods such as the finite element, finite volume and finite difference methods exist in a vast number of different variants, which also may be more or less associated with the PDE we want to solve, the type of numerical timestepping technique, the underlying (structured or unstructured) numerical grid and so on. Other variation points deal with computational simplifications of the problem: number of dimensions, Cartesian versus cylindrical versus spherical versus curvilinear coordinate system, symmetries (e.g., axio-symmetric), the actual size of the problem (i.e., the number of data points), and sequential versus parallel. In the traditional development of numerical software we make our pick of each of these alternatives. Of course the choices are well motivated by the problem we want to solve. To some extent, we may also decide for certain variations. Normally we will fully parameterise the data set size. We may choose to cater for 2D and 3D. If the circumstances are special we