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the equation is $$ \left\{\begin{array}{l} -\nabla \cdot \mathbb{T}(\mathbf{u}, p)=\mathbf{f} \text { in } \Omega, \\ \nabla \cdot \mathbf{u}=0 \text { in } \Omega, \\ \mathbf{u}=\mathbf{g} \text { on } \partial \Omega . \end{array}\right. $$ where $$ \mathbf{u}(x, y)=\left(u_1, u_2\right)^t, \mathbf{g}(x, y)=\left(g_1, g_2\right)^t, \mathbf{f}(x, y)=\left(f_1, f_2\right)^t . $$

  • The stress tensor $\mathbb{T}(\mathbf{u}, p)$ is defined as $$ \mathbb{T}(\mathbf{u}, p)=2 \nu \mathbb{D}(\mathbf{u})-p \mathbb{I} $$ where $\nu$ is the viscosity and the deformation tensor $$ \mathbb{D}(\mathbf{u})=\frac{1}{2}\left(\nabla \mathbf{u}+(\nabla \mathbf{u})^t\right) $$ The procedure of solving this in Taylor hood P2P1 FEM method is
  • Generate the mesh information matrices $P$ and $T$.
  • Assemble the stiffness matrix $A$ by using Algorithm I. (We will choose Algorithm I-3 in class)
  • Assemble the load vector $\vec{b}$ by using Algorithm II. (We will choose Algorithm II-3 in class)
  • Deal with the Dirichlet boundary condition by using Algorithm III-3.
  • Fix the pressure at one point in the domain $\Omega$.
  • Solve $A \vec{X}=\vec{b}$ for $\vec{X}$ by using a direct or iterative method. P stores the coordinate of nodes and T stores the index of nodes of each element, here I don't know how to fix the pressure at one point in the domain after Dealing with the Dirichlet boundary condition of $u$, is the point on which we want to fix the pressure must lie in the interior of the domain? how to do this in matlab?can any one help me ,thank you very much
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  • $\begingroup$ That looks a lot like a textbook question. $\endgroup$
    – Dan Doe
    Dec 21, 2022 at 10:32

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The Taylor Hood $(P_2,P_1)$ elements impose a continuous piecewise linear discrete pressure space in the Galerkin approximation.

So the discrete system of equations will have an unknown for pressure at each node of a triangulation, with matrix assembly treating the pressure as linearly interpolated among those nodes on a given (triangle) element.

One can add to the discrete system an equation assigning a value of the pressure unknown at any interior node. This extra condition will make the solution unique. There are other ways to impose an extra condition, but this would be consistent with the instructions for your class project.

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