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The Q2-P1 element is one of the popular finite elements for incompressible flow problems in the mathematics community. For this element, the velocity field is approximated using bi-quadratic shape functions and the pressure field using linear shape functions. What makes this element pretty odd is the fact that the approximation for the pressure field is only linear but not bi-linear, i.e. the function space consists of $1, \xi_1$ and $\xi_2$ only but not the product term, $\xi_1 \, \xi_2$.

I have two questions on this element.

  1. Where are the pressure DOFs stored, if not at the nodes?
  2. What is the correct way of interpolating the pressure field for numerical integration and also for plotting?

Thank you in advance! enter image description here

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The pressure is a discontinuous function, so you can't store nodal values at the vertices of the cells -- because then all neighboring cells would have the same value. Rather, if you want, you can pick any three points in the interior of the cell. In practice, people often choose three of the four vertices but consider these points logically part of the "interior" of the cell, rather than a vertex, to indicate that neighboring cells might have different values at these locations.

If you choose the three points in the cell to represent the pressure, then you also know how to interpolate an arbitrary function onto this space. For plotting, a common way is to recognize that a $P_1$ function on a cell is also a $Q_1$ function. That is, if you define the function based on the values at three vertices, compute what the value at the fourth vertex is and call it a $Q_1$ function that you can then output in whatever format you want.

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  • $\begingroup$ Thank you, Wolfgang Bangerth! If I understand correctly, $P_1$ space is a subset of the $Q_1$ space. Since there are three values internal to the element and four corners for the cell, the mapping is not unique. I am wondering if there is a mapping that is widely-used. $\endgroup$
    – Chenna K
    Jul 21 '20 at 20:56
  • $\begingroup$ First, just to be clear we understand each other: The element uses a continuous velocity space but a discontinuous pressure space. A common notation is to say that you are using the $Q_2 \times P_{-1}$ element, where the negative sign indicates the discontinuity. $\endgroup$ Jul 21 '20 at 23:49
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    $\begingroup$ Second: The basis is not unique for any other discontinuous element either. In fact, that is often even the case for continuous elements: Think of the $Q_3$ element, which has two degrees of freedom on each edge. Where exactly you place them is not canonic, and you can do whatever you want. For elements that are completely discontinuous, you can take any set of points in the (logical) interior of the reference cell. But you don't have to choose interpolation points: You could also use the representation $a+bx+cy$, for example -- except if you do it this way, you get an ill-conditioned basis. $\endgroup$ Jul 21 '20 at 23:51
  • $\begingroup$ I think, for the compatible iso-parametric elements, DOF naming convention is straightforward: place the DOFs at the nodes. For the $Q_3$ element, I would place the DOFs on the edge at the nodes on the edge. I don't think that there is any ambiguity here. The problem is with non-compatible shapes; using $P1$ on the Q1 element becomes problematic. I wonder what made the researchers create such an odd element. $\endgroup$
    – Chenna K
    Jul 22 '20 at 21:18
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    $\begingroup$ As for notation: $Q_2\times Q_1$ and $Q_2 \times Q_{-1}$ are different in that the former has a continuous pressure space whereas the latter does not (but is also not stable). For $Q_1\times P_{\pm 1}$: Both are understandable to the expert as discontinuous pressure spaces because there can not be a $P_1$ space that is continuous on quadrilaterals. So even if you write $P_1$ in this context, it is understood to be a discontinuous space. $\endgroup$ Jul 22 '20 at 23:58

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