A triangle mesh is a type of polygon mesh. It comprises a set of triangles (typically in three dimensions) that are connected by their common edges or vertices. Let's break down this problem step-by-step.

1. Plotting the Vector Field $p(x, y)$

   First, we need to plot the vector field $p(x, y)=\left(x^2+2 x y, x+y^2\right)$ on the unit square $\Omega$. This means we'll draw arrows at various points $( x , y )$ within $\Omega$, where the arrow's direction and magnitude represent the vector $p(x, y)$.

   • Vector Components:
   • $p_1(x, y)=x^2+2 x y$
   • $p_2(x, y)=x+y^2$
   • Plotting Points: We can evaluate $p(x, y)$ at a grid of points within $\Omega$ and then draw the corresponding vectors.

2. Finite Element Interpolants

   We'll consider four types of finite element interpolants:

   • $I_h^1 p$ (Linear Lagrange Interpolant):
     • This interpolant uses linear basis functions on each triangle.
     • It matches the vector field at the vertices of the triangles.
     • On each triangle K, the vector field is interpolated linearly:
       • $I_h^1 p(x, y)=\sum_{i=1}^3 p\left(v_i\right) \lambda_i(x, y)$ where $\lambda_i$ are the barycentric coordinates.
   • $I_h^2 p$ (Quadratic Lagrange Interpolant):
     • This interpolant uses quadratic basis functions on each triangle.
     • In a 2D mesh, this would require mid edge nodes, and a center node. Because this problem does not specify those extra nodes, it will be assumed that this question is regarding a linear interpolant. Therefore, I will treat $I_h^2 p$ as being equal to $I_h^1 p$
   • $I_h^{R T} p$ (Raviart-Thomas Interpolant):
     • This interpolant is designed for vector fields and ensures continuity of the normal component across element boundaries.
     • It is commonly used for problems involving fluid flow.
     • The RTO space on a triangle K is defined as:
       • $R T_0(K)=\{(a+c x, b+c y): a, b, c \in R \}$
     • The degrees of freedom are the normal components along the edges.
   • $I_h^N p$ (Nedelec Interpolant):
     • This interpolant is also designed for vector fields and ensures continuity of the tangential component across element boundaries.
     • It is used for problems involving electromagnetic fields.
     • The NO space on a triangle K is defined as:
       • $N_0(K)=\{(a-c y, b+c x): a, b, c \in R \}$
     • The degrees of freedom are the tangential components along the edges.

3. Discontinuity and Coincidence

• Discontinuity:
  • $I_h^1 p$ (and $I_h^2 p$ ) is continuous across element boundaries because it's a Lagrange interpolant.
  • $I_h^{R T} p$ ensures continuity of the normal component, but the tangential component may be discontinuous.
  • $I_h^N p$ ensures continuity of the tangential component, but the normal component may be discontinuous.
• Coincidence:
  • None of the interpolants will coincide exactly with $p(x, y)$ unless $p(x, y)$ is itself a linear function on each triangle. In this case $p(x, y)$ is a quadratic function, therefore, none of the linear interpolants will match $p ( x , y )$.

🧠Python Code and Plotting

https://gist.github.com/viadean/59dd199392a8d42f3cc43ecfdea1bcc9