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There's a vast amount of literature about the numerical solution of partial differential equations. But in none of my books or lectures did I find anything about coupled systems. Internet and library research also did not lead to any introductory material.

Could anyone recommend a textbook or a paper with an introduction on how to treat coupled PDEs numerically? Particularly for concentrations and potentials in electrochemistry.

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  • $\begingroup$ With coupled problems, one important distinction is whether the equations are 1) defined on distinct domains and coupled through their interfaces or 2) defined on overlapping domains so that the degrees of freedom are (locally) coupled throughout the entire domain. How are your particular equations coupled? $\endgroup$
    – Paul
    Commented Oct 3, 2014 at 14:54
  • $\begingroup$ Another interesting reference is Automated Solution of Differential Equations by the Finite Element Method that is the book of the FEniCS project. $\endgroup$
    – nicoguaro ♦
    Commented Oct 3, 2014 at 23:35

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Finite element methods for coupled systems are treated for example in Chapter 18 of The Finite Element Method: Its Basis and Fundamentals by Zienkiewicz, Taylor and Zhu. It might also be useful to search for literature on particular coupled systems (presumably there's a specific one you are interested in) such as those modeling fluid-structure interaction or electromechanical systems (e.g., in piezoelectricity or cardiophysics).

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There are of course many aspects to solving coupled problems (discretizations, solvers, ...). Assuming you are interested in solvers and preconditioners, then lectures 37 and 38 at http://www.math.tamu.edu/~bangerth/videos.html may have something for you. (Disclaimer: these are my own lectures.)

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