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I have a 2D Poisson problem on a circle domain with Robin boundary conditions: $$-\Delta u = f \text{ on } \Omega$$ $$u + \varepsilon \partial_n u = 0 \text{ on }\partial\Omega$$ I want to solve this problem with piecewise linear finite elements and would like to compare my implementation with the theoretically expected error decay.

In my NumPDE course I have seen a theoretical result, if I would instead consider the Poisson problem with Dirichlet boundary conditions. There (under the assumption of a shape regular traingulation and under a regularity assumption on the solution) one obtains that the $H^1$ error is $\mathcal{O}(h)$.

My question now is, whether one would also expect this error decay theoretically for my example problem. I can see two problems that I do not really know how to deal with:

  1. I have Robin boundary conditions instead of Dirichlet boundary conditions. Does this change the estimate?
  2. I have a circular domain. In my lecture it was not stated clearly whether the convergence result above only holds on polygonal or also on circular domains. Is this a further problem?

I would be very thankful, if anybody can comment on these questions or can even provide a reference with the adequate result in my situation.

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I'll assume you're using a continous piecewise polynomial space $V_h$. For question 1, this should be fine, the result follows from the coercivity, and continuity of the bilinear form $a_h(u_h,v_h)$ (in the H^1 norm), and the Galerkin orthogonality property: $$a_h(u - u_h,v_h) =0\quad\forall v_h\in V_h,$$ where $u$ is the PDE solution, and $u_h$ is the finite element solution. From this it is straightforward to prove a Cea's lemma type result: $$\|u-u_h\|\le C\inf_{v_h\in V_h}\|u-v_h\|,$$ then the O(h) approximation result follows from standard polynomial approximation theory.

For 2, it is okay to work on a disc, but you may be commiting what is called a variational crime (since you can't represent a disk exactly with a polygonal mesh). Typically this results in limiting the convergence rates to $O(h^2)$, but this is good enough to cover your O(h) approximation.

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