I'm currently developing simulations for coupled, nonlinear, multi-region systems. Basically, I use the Finite Element Method (FEM) to model each physical quantity in each region. The obtained matrices and the coupling matrices between the quantities are combined into one large system matrix. From this matrix plus the derivatives of the nonlinear functions, a Jacobi matrix is generated. The system is then finally solved by the Newton-Raphson method. This includes the inversion of the Jacobi matrix, which is either done iteratively or directly.

I do not have a specific question, but I observe that I'm running into problems over and over again. Bad convergence, instabilities, oscillations, to name some of them. It's proably because I have almost no theoretical background in the field. FEM itself is not the problem, but the solution of the equations in a coupled and nonlinear scenario.

I'm looking for some good material to get some basic theoretical understanding. What's a good introductory book or lecture about the topic? (I'm a graduated engineer, so at least the basic maths is - hopefully - not the problem.)

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    $\begingroup$ Since, as you say, you don't really have a specific question, it is difficult to provide very specific suggestions. However, I suggest taking a look at the notes from Felippa's nonlinear FEM class (colorado.edu/engineering/CAS/courses.d/NFEM.d/Home.html). You'll notice he has eight sets of notes specifically on solution methods for nonlinear problems. $\endgroup$ – Bill Greene May 31 '17 at 16:34

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