# How to implement pressure stabilization in matlab while solving steady stokes equation

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
• That looks a lot like a textbook question. Dec 21, 2022 at 10:32

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