I'm interested in suggestions for book references on the subject of numerical PDE and ODE, in particular, a rigorous analysis of such methods in a manner written for professional mathematicians. It does not have to be extremely comprehensive in the sense of listing hundreds or thousands of different methods, but I would be interested in something that at least covers most of the key concepts that guide modern techniques.
I think it would be proper to draw analogies to textbooks on numerical linear algebra, about which I am more familiar. I'm looking for something that is to stability and truncation errors in numerical differential equations as Higham's Accuracy and Stability of Numerical Algorithms is to stability and roundoff errors in numerical linear algebra, and something that discusses modern techniques in ODE and PDE the way that Golub and Van Loan's Matrix Computations discusses most of the main types of techniques for linear algebra.
I actually know very little about numerical ODE and PDE. I've been reading through some assortment of online notes, and I have the book Finite Difference Methods for Ordinary and Partial Differential equations by Randall LeVeque, which is a clear book but not in-depth enough for my purposes. As a more concrete example of the level I'm looking for, I would hope that any section on elliptic and parabolic equations assumes the reader has full familiarity with the theory of Sobolev spaces and their embeddings, and weak solutions for PDE, and uses results from that theory rather freely in deriving error estimates for finite elements, etc.