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I'm trying to understand what a mixed FE method is, and I was recommended this pdf: http://www.ima.umn.edu/~arnold/papers/mixed.pdf (Arnold, Mixed Finite Element Methods for Elliptic Problems).

On first page the author states the motivation as 'treating stress and displacement fields as primary variables'. However the equations are:

$$ \mathbb{A}\mathcal{S} = \mathcal{E}(\mathbf{u}) \\ \mathbf{div}\,[\mathcal{S}] = \mathbf{f} $$

The author says $\mathbb{A}$ is a SPD operator so first equation can be rewritten as:

$$ \mathcal{S} = \mathbb{C}\,\mathcal{E}(\mathbf{u}) $$

However we can substitute this into the 2nd of the first two equations and get:

$$ \mathbf{div}\,[\mathbb{C}\mathcal{E}(\mathbf{u})] = \mathbf{f} $$

and so $\mathcal{S}$ is eliminated, and we don't need to consider two primary variables (all we need to do is to solve for $\mathbf{u})$.

So what's a better motivation for using mixed-method or what is wrong with the above reasoning?

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  • $\begingroup$ There are different motivations for mixed methods, depending on the context, but this particular argument comes from applications where the stress is actually something you are interested in and which in the second approach -- which is perfectly valid and actually the standard formulation of elasticity -- would have to be computed in a separate step from the displacement. $\endgroup$ – Christian Clason Nov 17 '17 at 23:20
  • $\begingroup$ You lose accuracy by post processing the stress after computing the displacement. You can get better accuracy by making stress a primary variable to be solved, but you'll need a difference space with sufficient properties to achieve this goal. Also, some mixed methods can avoid locking phenomenon. In other PDE's, some mixed methods produce more stable results than their non-mixed counterparts. $\endgroup$ – Paul Nov 18 '17 at 4:09
  • $\begingroup$ Are you only interested in this application (splitting of elasticity system) or others applications as well? $\endgroup$ – knl Nov 19 '17 at 17:42
  • $\begingroup$ @knl Well at the moment I'm not interested in applications but the general concept. If the general concept is better explained through a different example, I'm okay with that. $\endgroup$ – Fi Zixer Nov 19 '17 at 17:45
  • $\begingroup$ Like Christian Clason mentioned, there are different motivations depending on the context. I think the best motivation comes from natural saddle point problems, like Stokes or elasticity with contact. $\endgroup$ – knl Nov 20 '17 at 13:35
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Mix formulation is conceptually more difficult, and one has to choose approximation spaces appropriately to make discrete system stable.

Matrix emerging from mix formulation is very sparse but has size bigger compared to equivalent single filed formulation. However, very often mix and classical formation has the comparable number of non-zero entries. Properties of matrix emerging from mix formulation are not as good as fro classical formulation. Very often one need solver tailored to the problem.

For elastic problems, locking is an argument for mix formulation, however, range of methods/remedies are available which removing locking with small cost, for example, efficient and straightforward B-bar method in plasticity,

So overall you have to have good argument why you like to apply mix elements.

For me, mix-formulation has four attractive adventages;

1) You have error evaluator build in the formulation of mix element. To implement classical element, you need half an hour, but to implement good error estimator you need a month.

2) Stresses or in general fluxes has a higher order of convergence. Those are usually primary variable of interest for an engineer, evaluating the strength of the structure.

3) For nonlinear problems, you can decouple nonlinear (e.g. constitutive/physical equation) and linear (e.g. momentum conservation) part of the equation. That enables you to build efficient, stable nonlinear solvers, for example LaTIn method.

4) You can capture irregular solutions, abrupt changes, shock easier, for example in acoustics or unsaturated flows when material properties changes sharply.

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  • $\begingroup$ Quite great answer! $\endgroup$ – Wenjin Xing Nov 19 '17 at 7:10
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The primary motivation for mixed methods is that you want both the displacement and the stress to be primary variables.

If you substitute the one equation into the other, then you only have the displacement as the primary variable. You can then, at a later time, compute the stress from the displacement, but this post processing step usually leads to lower accuracy in the computed stress than in the computed displacement.

On the other hand, if you keep both of these variables as primary variables, you can choose the approximation order of the two independently (within limits), and you can guarantee that the computed stress is at least as accurate as the computed displacement, and in general the stress will be more accurate this way than if you computed it as a postprocessing step from the displacement.

(If you now ask why anyone would need the computed stress to be accurate: Think about cases where you need to know whether a material will break -- i.e., whether the stress exceeds a certain limit -- or where the stress determines the behavior of the material -- for example, in plasticity.)

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