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I was reading the book: The Finite Element Method: Theory, Implementation, and Applications by Mats Larson and Fredrik Bengzon, in page 140 of this book they say this:

"The Lax-Milgram Lemma is one of the most important theorems in applied mathematics"

but, why?!... to answer this we recall that many boundary-value problems for ordinary and partial differential equations can be posed in the following abstract variational form: Find $u \in V$ such that

\begin{equation} a(u,v) = l(v), \ \ \forall v \in V \tag{1} \label{vF} \end{equation}

where $V$ is a normed space. For example the weak formulation in the finite element method for Poisson's equation.

The Lax-Milgram Lemma gives sufficient conditions for the existence and uniqueness of the solution for (\ref{vF}).

then I wonder,$\mathbf{what \ are \ the \ most \ important \ theorems \ in \ computational \ science?}\tag{*}\label{Q} $

I've thought about the Picard's existence theorem that gives conditions under which an initial value problem has a unique solution.

My question is given by (\ref{Q}). Thanks!

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    $\begingroup$ This might sound crazy but I think the demonstration of the convergence of Taylor series is a critical one. So many schemes and methods can be derived using Taylor series. $\endgroup$ – BlaB Jan 8 at 21:56
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    $\begingroup$ I think narrowing/elaborating on this question is pretty wise. What specific region of computational science? Do you just want the broad theorems computational science works off of, because then you get a lot of existence and uniqueness proofs? Do you want something more focused on theorem that allow for various algorithms depend on? $\endgroup$ – EMP Jan 8 at 23:03
  • $\begingroup$ @EMP I'm not interested in a specific area of computational science, I thought that maybe we can try to have a list of important theorems per area. For example BlaB mentioned Taylor's Theorem, which is an important theorem from analysis, that is usefull to derive finite difference formulas. $\endgroup$ – tnt235711 Jan 8 at 23:25
  • $\begingroup$ Computational science is about computing, and what actually works well in practice, not about theorems. $\endgroup$ – Mark L. Stone Jan 10 at 20:01
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You'll get everyone to give different answers to this question, and maybe that's alright. Here are some of my favorite ones:

  • Taylor's theorem that a function (of sufficient smoothness) equals its Taylor expansion plus a remainder term.

  • One can consider the Bramble-Hilbert lemma as a variation of Taylor's theorem, but it has different applications and is important of its own.

  • The Lax-Milgram lemma about the existence and uniqueness of solutions of variational problems.

  • The Babuska-Brezzi (LBB) condition about stable solutions to saddle point problems.

  • The fact that Gauss quadrature works, i.e., that we can find locations and weights for quadrature rules with $n$ points that lead to $O(h^{2n-1})$ convergence.

  • The theorem that proves that Newton's method for finding a minimum (or root of a function) converges if one starts close enough to the solution, and that connects the radius of convergence to the size of higher derivatives.

  • The theorems that Krylov subspaces are useful in devising iterative solvers for linear systems, first demonstrated in proving the convergence of the Conjugate Gradient method.

Each of these have been used in much follow-up research that has generalized and adapted them to new situations.

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    $\begingroup$ Thank you so much Prof. Wolfgang Bangerth, this is the kind of answer that I had in mind. Believe it or not, I was expecting your answer...I'd like to know the rest of your favorite theorems. $\endgroup$ – tnt235711 Jan 9 at 2:29
  • $\begingroup$ I might add the Babuska-Brezzi condition for saddle point problems. $\endgroup$ – Wolfgang Bangerth Jan 9 at 4:58
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    $\begingroup$ Also the crazy fact that Gauss quadrature works. I guess I should just augment my answer. $\endgroup$ – Wolfgang Bangerth Jan 9 at 4:59
  • $\begingroup$ nothing about adjoints? I feel like that's pretty important in CS, but YMMV $\endgroup$ – EMP Jan 9 at 21:06
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    $\begingroup$ @EMP: Feel free to propose your own list :-) I'm definitely seeing things through my own lense. $\endgroup$ – Wolfgang Bangerth Jan 10 at 0:54
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Since this is a question with pretty subjective answers, I'll add a couple to Prof. Bangerth's very good list.

  • the theorem of adjoint/dual operators and spaces is pretty crucial to Computational Science. We know that dual-consistent discretizations of the PDEs can obtain superconvergence which is a nice property. But I think the more commonly used outcomes are the continuous and discrete adjoint (mainly the discrete) allow for an output based error estimate criteria for mesh refinement and efficient gradient computation. These are used in most of computational science to my knowledge.
  • the theorem and algorithms behind multigrid/multilevel solvers. Some of the original papers come from Brandt, and in CFD you can look into the Jameson/Mavriplis papers. Diskin (one of Brandt's students) has written a lot as well on the topic. Multigrid is just about the best and can be appplied to both linear and nonlinear solvers and makes large scale problems possible that were previously impossible.
  • Lastly, I'll just throw in vector calculus here, as basically every PDE we simulate uses vector calculus and divergence theorem.
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