On object-oriented frameworks and coordinate free formulations of PDEs

An object-oriented (OO) framework for Partial Differential Equations (PDEs) provides software abstractions for numerical simulation of PDEs. The design of such frameworks is not trivial, and the outcome of the design is highly dependent on which mathematical abstractions one chooses to support. In this paper, coordinate free abstractions for PDEs are advocated. The coordinate free formulation of a PDE hides the underlying coordinate system. Therefore, software based on these concepts has the prospect of being more modular, since the PDE formulation is separated from the representation of the coordinates. Use of coordinate free methods in two independent OO frameworks are presented, in order to exemplify the viability of the concepts. The described applications simulate seismic waves for various classes of rock models and the incompressible Navier-Stokes equations on curvi-linear grids, respectively. In both cases, the possibility to express the equations in a domain independent fashion is crucial. Similarities and differences between the two coordinate free frameworks are discussed. A number of places where such frameworks should be designed for modification is identified. This identification is of interest both for framework developers and for tentative framework users.