Block operators and spectral discretizations

Every student of numerical linear algebra is familiar with block matrices and vectors. The same ideas can be applied to the continuous analogues of operators, functions, and functionals. It is shown here how the explicit consideration of block structures at the continuous level can be a useful tool. In particular, block operator diagrams lead to templates for spectral discretization of differential and integral equation boundary-value problems in one space dimension by the rectangular differentiation, identity, and integration matrices introduced recently by Driscoll and Hale. The templates are so simple that we are able to present them as executable Matlab codes just a few lines long, developing ideas through a sequence of 12 increasingly advanced examples. The notion of the rectangular shape of a linear operator is made mathematically precise by the theory of Fredholm operators and their indices, and the block operator formulations apply to nonlinear problems too. We propose the convention of representing nonlinear blocks as shaded. At each step of a Newton iteration for a nonlinear problem, the structure is linearized and the blocks become unshaded, representing Fréchet derivative operators, square or rectangular.