I am using the finite element method and need to find the errors associated with each of my elements. I am looking for help to find the error on the edges of the triangle, preferably by hand and not code.

For this I know $R_K(u_h)=f+\Delta u_h=1$ as f=1 and $u_h$ is linear.

Now for the edges we know the following, $R_E(u_h)=-\underline{n}_E\cdot\nabla u_h \ \text{for} \ E\in\epsilon_\omega$ where $\omega$ is our domain, noting that we have Dirichlet conditions on the boundaries s.t. $u(0)=0$. Which is where I become stuck.

Now I am considering the unit square with 4 equally spaced nodes, see the picture. After running the FEA code I get solutions of the form $u_h =( 0.0194, 0.0186, 0.0203, 0.0217)$, which is plotted below.

If someone can help me with finding $R_E(u_h)$ that would be great!

enter image description here

Solution of FEA


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