In my lecture I learnt that if I solve the Poisson problem with Dirichlet boundary conditions on a simplicial shape regular mesh with finite elements with polynomials of degree k, I'll obtain a convergence rate of $\mathcal{O}(h^k)$. For the $L^2$ error I gain one order and obtain $\mathcal{O}(h^{k+1})$. Question: Is this a general phenomenon or do such results only hold for this special case? In particular: Do I also get those orders for Neumann, Robin or Mixed Boundary Conditions and which requirements on the domain do I have? Furthermore, is there a good reference/overview where I can find similar results/related discussions?
2026-03-29 04:48:58.1774759738
Convergence rates for finite elements
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This result is observed to hold for a wide range of examples, however it is in general difficult to show (i.e. there are lots of examples where it holds, but there is no proof of the result). The typical approach for proving the result relies on being able to solve the dual problem. These lecture notes are a good place to start: https://www.google.com/url?sa=t&source=web&rct=j&url=https://people.maths.ox.ac.uk/suli/fem.pdf&ved=2ahUKEwitrdaY1ZL6AhXNSUEAHRWPD88QFnoECA8QAQ&usg=AOvVaw3cJrPxVtcAha8DUqqWf13L