Operators, functions, and functionals are combined in many problems of computational science in a fashion that has the same logical structure as is familiar for block matrices and vectors. It is proposed that the explicit consideration of such block structures at the continuous as opposed to discrete level can be a useful tool. In particular, block operator diagrams provide templates for spectral discretization by the rectangular differentiation, integration, and identity matrices introduced by Driscoll and Hale. The notion of the rectangular shape of a linear operator can be made rigorous by the theory of Fredholm operators and their indices, and the block operator formulations apply to nonlinear problems too, where the convention is proposed of representing nonlinear blocks as shaded. At each step of a Newton iteration, the structure is linearized and the blocks become unshaded, representing Fréchet derivative operators, square or rectangular. The use of block operator diagrams makes it possible to precisely specify discretizations of even rather complicated problems with just a few lines of pseudocode.
[Joint work with Nick Trefethen]
- Numerical Analysis Group Internal Seminar