Is their a book/paper about mixed finite element method for engineering students (non-math)?

Question

There are a lot of books about FEM, which are really friendly to engineering students. Through these books, we can know how to use shape/test functions based on the variational principle. But I'd like to solve the mixed formulation of Poisson equations (i.e. Darcy equation), which reads

$\mathbf{u}=-k\nabla h$

and

$\nabla \cdot \mathbf{u} = 0$,

where $\mathbf{u}$ is velocity, $k$ is a conductivity coefficient and $h$ is pressure.

There are a number of papers suggesting that the mixed FEM should be used to solve the equations insteads of regular FEM, e.g.

Garnadi, Agah D., and Corinna Bahriawati. "A mixed $ RT_0-P_0 $ Raviart-Thomas finite element implementation of Darcy Equation in GNU Octave." (2020).

These works were finished by mathematicians, which present a lot of difficult concepts, like functional space, and mathematic symbols. The equations were always presented in a simple way that is difficult to understand. Also, by reading other similar papers, I found that they prefer to prove their mixed FEM method is sound theoretically in math. Although some papers give MATLAB codes, I still got confused because I cannot understand how to perform mixed FEM.

Is there a book/paper friendly to engineering students? I mean, just tell me what is exact form of the test functions? how to deduce the final algebraic equations? how to address boundary conditions? and something like that.

Answer 1

I think, one should distinguish two goals here:

1. Understand, at a superficial level, mixed FEMs.
2. Only concerned about implementing them.

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I will start with the second use case, as I feel from your question that this is what you are after.

just tell me what is exact form of the test functions? how to deduce the final algebraic equations? how to address boundary conditions?

These are implementation details. I recommend you to look at the source code of established, high-quality FEM frameworks, such as deal.ii or FEniCS. You can find many detailed tutorials in deal.ii, which are at an accessible level and in which the emphasis is on the practical implementation. For instance, Tutorial 20, and the corresponding video, discusses exactly your problem.

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On the other hand, if you want to develop a new mixed formulation, you must have understanding of the mathematics, you cannot avoid it. Having an engineering background, I faced similar issues as you: function spaces everywhere. I recommend you to have a firm grasp on non-mixed formulations first (i.e. standard FEM); there are many good tutorials on them. Then make a connection with mixed FEMs by understanding the main difficulty of mixed formulations, i.e. the careful choice of the discrete spaces. Note that proving that a mixed formulation is stable is difficult, and research papers are not the good source to have an intuition. There are books on mixed FEMs, but they are not for completely novices. I am afraid there is no shortcut here, you won't find a book/tutorial like "Learn mixed methods in 24 hours".

Answer 2

The following paper is a good starting point.

• Arnold, D. N. (1990). Mixed finite element methods for elliptic problems. Computer methods in applied mechanics and engineering, 82(1-3), 281-300. Author copy: https://www-users.cse.umn.edu/~arnold/papers/mixed.pdf

It presents the general problem and why one would like to use one formulation or the other. Also, there is some discussion on how to choose stable mixed finite elements.

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I asked a friend this question and she answered the following.

If one knows FEM and wants to know Mixed FEM it is really difficult because there are plenty of combinations of basis functions that are unstable. The best shot is to introduce the LBB condition, which requires a functional analysis background. This is what Gatica tries in the following book

• Gatica, G. N. (2014). A simple introduction to the mixed finite element method. Theory and Applications. Springer Briefs in Mathematics. Springer, London.

There, the starting point is the Stokes equation and a naive choice of basis. Then the Raviart-Thomas element appears naturally.

From an implementation point of view, she suggests the following, besides Gatica's book.

• Bahriawati, C., & Carstensen, C. (2005). Three MATLAB implementations of the lowest-order Raviart-Thomas MFEM with a posteriori error control. Computational Methods in Applied Mathematics, 5(4), 333-361.