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I'm looking for some algorithm of a finite elements to solve the problem

Find $u,p\in H_0^1(\Omega)\times L^2(\Omega)$ such that

$-\nu\Delta u+\nabla p=f$ in $\Omega\subset\mathbb{R}^2$

$\nabla\cdot u=0$ in $\Omega$

$u=0$ in $\partial\Omega$

where $\Omega$ is a bounded subset o $\mathbb{R}^2$, and we have a typical finite element mesh of $\Omega$ composed by triangles (its union is all $\Omega$, are not overlaped...), that is, the simplest and classic Stokes equations with lowest finite elements, nothings strange here.

I like to program a finite element code (just for practice) where the velocity $u_h\in H_h$ (continuous over $\Omega$) be approximate by piecewise linear elements over triangles and $p_h\in Q_h$ constants over each triangle of the mesh. $u_h$ and $p_h$ (discontinuous over $\Omega$) are the approximations by the finite elements method.

Do you know some easy finite method for practice? Some book, paper, pdf... with friendy notation.

I know that the scheme: find $(u_h,p)\in H_h\times Q_h$

$(\nu\nabla u,\nabla v)-(\nabla\nabla u,q)+(\nabla\nabla v,p)=(f,v)$ for all $v\in H_h$ and $q\in Q$

is not well possed. For that reason I'm looking for some well-posed scheme for practice the programming of the method.

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I think the method given in

Hughes, T. J. R. and Franca, L. P. [1987]. A new finite element formulation for computational fluid dynamics: VII. The Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces.

is stable for $P_1-P_0$ but the programming is really tricky since you need to add jump terms between elements. I'm quite positive that this is the only stable $P_1-P_0$ method with continuous velocity.

If you want an easy method to program, choose one from the following list. We have used all of them as a project work in our university for students who wanted to practice solving Stokes equation with finite elements.

  • MINI element
  • Stabilized $P_1-P_1$ element (continous pressure)
  • Taylor-Hood, i.e. $P_2-P_1$ element

All of them are described in the book The Finite Element Method: Theory, Implementation, and Applications in the chapter Fluid Mechanics.

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