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The advent of high-speed computers has made it possible for the first time to calculate values from models accurately and rapidly. Researchers and engineers thus have a crucial means of using numerical results to modify and adapt arguments and experiments along the way. Every facet of technical and industrial activity has been affected by these developments. The objective of the present work is to compile the mathematical knowledge required by researchers in mechanics, physics, engineering, chemistry and other branches of application of mathematics for the theoretical and numerical resolution of physical models on computers. Since the publication in 1924 of the "Methoden der mathematischen Physik" by Courant and Hilbert, there has been no other comprehensive and up-to-date publication presenting the mathematical tools needed in applications of mathematics in directly implementable form.



Chapter X. Mixed Problems and the Tricomi Equation

Numerous problems of mathematical physics come down to solving a Dirichlet problem for a linear elliptic equation. Variational methods, in the framework of Sobolev spaces, are then particularly well adapted. Many examples have been illustrated in the previous chapters.
Robert Dautray, Jacques-Louis Lions

Chapter XI. Integral Equations

Numerous stationary problems (see Chaps. IA and IB) are of the following type: find u satisfying
$$ Au = f $$
, where A is a linear operator (continuous or not) in F, where F is a topological vector space and where f is given in F.
Robert Dautray, Jacques-Louis Lions

Chapter XII. Numerical Methods for Stationary Problems

In the preceding chapters we have reviewed the results concerning the existence and uniqueness for stationary elliptic problems.
Robert Dautray, Jacques-Louis Lions

Chapter XIII. Approximation of Integral Equations by Finite Elements. Error Analysis

We present here the approximation by finite elements of integral equations on a surface Γ of ℝ3, associated with the Laplace equation in ℝ3. The extension of these techniques to other integral equations introduced in Chap. XI presents no new difficulties.
Robert Dautray, Jacques-Louis Lions


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