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I have seen finite difference methods for fluid equations (Stokes and Navier--Stokes) that solve a pressure problem first and then a fluid problem. That is, although they solve two different problems, they are two smaller problems, which from a computational point of view is a tremendous advantage compared to solving only one problem.

In the Finite Element methods, I have seen that they solve only one problem, for example in Taylor--Hood. Why is it not a good idea to break the problem down into two (one for pressure and one for velocity) in Finite Elements? Or maybe it's possible, but not a popular method.

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    $\begingroup$ Are you interested in the time-dependent or the stationary Stokes equations? $\endgroup$ Commented Oct 9, 2022 at 23:57
  • $\begingroup$ Hello @WolfgangBangerth, I'm interested in the stationary Stokes equation in its simpler version, like simweb.iwr.uni-heidelberg.de/~darndt/files/doxygen/deal.II/… (with laplacian instead of epsilon, constant eta, and Dirichlet+Neumann boundary conditions). Do you know if there are fem methods that solve the pressure and velocity problem separately? Something like decoupling the problem in two? Because all the methods I've seen solve a single problem with unknown velocity and pressure together $\endgroup$
    – yemino
    Commented Oct 11, 2022 at 1:28
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    $\begingroup$ I think one can consider the Uzawa iteration such a method. You may also want to look at the step-22 tutorial of deal.II as an example of where we first solve for the pressure, and then for the velocity. $\endgroup$ Commented Oct 11, 2022 at 3:30

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Such splitting methods as you described, known as projection or fractional-step methods, are indeed available for FEM. The Characteristic Based Split (CBS) method is quite popular among such methods, see Zienkiewicz et al. for the details.

For such methods, the clear advantage is (i) in smaller matrix systems when compared to the coupled velocity-pressure formulation and (ii) avoidance of the saddle-point problem which occurs in the coupled formulation. We can also use equal-order interpolation in some cases. However, the disadvantage of splitting methods is their reduced accuracy and stability; the moment we split, we lose accuracy. Another disadvantage is the loss of numerical damping. Refer to our recent paper which studies these issues comprehensively.

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