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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.

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    $\begingroup$ A warning: for nonlinear systems of hyperbolic PDEs, none of the state-of-the-art methods in actual use is provably convergent (we don't even have the tools to prove well-posedness of the problems, typically). So you have to choose between proving things for toy problems/methods or learning about practically significant problems/methods. $\endgroup$ – David Ketcheson Mar 23 '13 at 13:12
  • $\begingroup$ @DavidKetcheson, yes, it's a little unfortunate. My general interests lie in trying to develop theory for computational methods, however quixotic that may be. $\endgroup$ – Christopher A. Wong Mar 23 '13 at 21:09
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You won't find one reference systematically covering the analysis of all the important methods for PDE. The field of discretization techniques for PDE is at least an order of magnitude larger than either topic you mentioned above. For any methods involving implicit solves, studying discretizations without also considering solution methods (e.g., associated multigrid methods) is a tried and true way to paint yourself into the "hopelessly impractical" corner.

Presumably you are familiar with Brenner and Scott, The Mathematical Theory of Finite Element Methods. It is a graduate level text, and although it has its share of introductory matter, you can quickly get to the important results.

For a posteriori error analysis in FEM, a good source is the review paper, Ainsworth and Oden, A posteriori error estimation in finite element analysis, 1997.

For finite volume methods, you might like the Acta Numerica paper Morton and Sonar, Finite volume methods for hyperbolic conservation laws, 2007. As Acta Numerica papers go, this is not very highly cited. I suspect that is partly because LeVeque's book is very good and because most practitioners that have not used his book are familiar with many of the original sources. Although I am not familiar with it, you might also look at Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws.

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I second Jed's point about the importance of considering solvers at the same time as discretizations. This is something "purer" mathematicians sometimes fail to do, much to their detriment, as they are solving the wrong problem. Things like the block structure, sparsity pattern, and ability to build preconditioners tends to be much more important than simple things like number of degrees of freedom/mesh size.

Brezzi & Fortin - "Mixed and Hybrid Finite Element Methods" covers material complementary to Brenner and Scott. It's out of print though and people really hang onto their copies, so if you don't want to pay several hundred dollars you'd probably have to borrow it from your library.

The series of papers by Rannacher et al in the early 2000's such as "An Optimal Control Approach to A Posteriori Error Estimation in Finite Element Methods" provides a deeper and more widely applicable understanding of a posteriori error estimation than what is explained in Ainsworth and Oden's book (in my opinion).

Sobolev spaces are not the be-all-end-all function spaces for PDE's, though you might get that impression reading introductory graduate books such as Evans. Besov spaces are more general and quite nice, and force you to think about how and why certain function spaces are constructed by controlling basic building blocks to provide constraints on oscillation, integrability, and multiscale structure. A nice "philosophical" article on the subject of function spaces broadly is Terry Tao's post here. Triebel's book (mainly about Besov spaces), "Theory of Function Spaces II", is great! There is a deep connection between Besov spaces and wavelets, so DeVore's very readable article on wavelets is useful.

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  • $\begingroup$ Actually, Springer has recently made the Brezzi & Fortin book available again (as print on demand). You might want to hold out, though, as there is apparently an updated version coming out in May. $\endgroup$ – Christian Clason Mar 24 '13 at 10:01
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In addition to Jed's great recommendations (I can personally vouch for Brenner+Scott as a great intro finite elements book), an excellent book for the numerical solution of ODEs is Butcher:

http://books.google.com/books/about/Numerical_Methods_for_Ordinary_Different.html?id=opd2NkBmMxsC

That was my bible for a good while, until my university library had it recalled.

Also you might find Ern+Guermond to be a valuable book, if you already are comfortable with the delicate mathematics

http://books.google.com/books/about/Theory_and_Practice_of_Finite_Elements.html?id=CCjm79FbJbcC

Having read a few papers from Ern+Guermond I can say they definitely lean towards heavy formalism. The chapters are more or less self contained modulo some notation that you might have to flip around to get the definition of.

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  • $\begingroup$ Thanks for your suggestions. The book by Ern & Guermond has the sort of flavor I'm oriented towards, being a mathematician. $\endgroup$ – Christopher A. Wong Mar 23 '13 at 10:26
  • $\begingroup$ In this case you might also be interested in DiPietro' and Ern's book on discontinuous Galerkin methods. $\endgroup$ – Christian Clason Mar 24 '13 at 9:57
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For PDEs, a book with a similar functional-analytic flavor as Ern and Guermond is D. Braess, Finite Elements, Cambridge University Press, 2007. Being a textbook rather than a research monograph, it is more accessible, although less comprehensive. On the other hand, it also discusses applications (mainly in elasticity).

Regarding ODEs, I believe the bible is still the three-volume work by Hairer and Wanner (Solving ODEs I, Solving ODEs II and Geometric Numerical Integration).

Finally, don't overlook the many excellent lecture notes available on the internet.

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