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I wanted to learn how to implement a code for the Stokes and Navier--Stokes equations 2D/3D. I already know how to implement it when the elements are triangles or tetrahedral.

Do exists finite element schemes using only square/cube elements for the pressure and velocity? Can you suggest me some paper or book with the method?

Thanks in advance.

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    $\begingroup$ If you know how to implement tria/tetra elements, then it is straightforward to implement quad/hex elements. You need to modifications to account for the change in the number of nodes and implement appropriate subroutines for the basis functions and quadrature points.. You can refer to my repository for the details. $\endgroup$
    – Chenna K
    Jul 19, 2021 at 22:01
  • $\begingroup$ Thanks @Chenna K. But, for example, in the case of a mesh composed by triangles T, we build the set of base functions P1(T) (set of polynomial of degree less that one on a triangle T of the mesh) such that the value of one element in one node is 1 and 0 in the other nodes, how can build the basis functions in squares (for example)? I think that it is the only big difference. A more specific question: in a reference triangle T, the base is {1-x-y,x,y}, but which is the base on a reference square? $\endgroup$
    – yemino
    Jul 19, 2021 at 22:41
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    $\begingroup$ Basis functions for quad/hex elements are formed by taking the tensor product of basis functions in 1D. Refer to this page for the details. $\endgroup$
    – Chenna K
    Jul 20, 2021 at 0:10

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Yes. 2D/3D methods with quad/hex only elements exist. Gresho and Sani's book: "Gresho, P. M. (Philip M.), Robert L. Sani, and M. S. Engelman. Incompressible Flow and the Finite Element Method : Advection-Diffusion and Isothermal Laminar Flow; P.M. Gresho, R.L. Sani, in Collaboration with M.S. Engelman. Chichester;: Wiley, 1998", has a table of the stable elements for the mixed-Galerkin method that were well known at the time of its publication and the problems with other elements. The whole scheme is laid out therein. I can only give you the link to the basic source since I'm working from home due to COVID-19, otherwise I'd give you a summary of the table and a tighter reference. My institution (UT Austin) has an electronic copy available to me, but you might be able to find it via yours. It appears to have been split into two volumes now. I found my copy used many years ago, but I think, looking at the table of contents, that Volume Two might be what you're looking for.

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