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In a smooth domain in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$ let's consider $-\Delta u = f$ with $u=g$ on a part of the boundary and $\partial_\nu u = w$ on another part of the boundary, which is far away from the previous one, to avoid compatibility conditions.

Now I want to solve this problem employing linear FEM and the easiest way to do it is by solving $(\nabla u_h, \nabla v_h)_{\Omega_h}=(f_h, v_h)_{\Omega_h}+(w_h,v_h)_{\partial\Omega_h}$, $u_h=g_h$ on ${\partial \Omega_h}$. Here $g_h, w_h$ are the nodal interpolants of $g,w$ and $\Omega_h$ is a polyhedral approximation of the original domain, having external nodes on $\partial \Omega$, and $f_h$ is an appropriate extension of $f$.

What is known about the convergence properties of this method? A literature research was surprisingly inconclusive, yet this is the way in which some standard PDE solvers (e.g. Fenics) work.

I would expect convergence of optimal order since the FEM used are linear, and so is the approximation of geometry.


Note: cross posted on MO

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Well, first, switching between Dirichlet and Neumann boundary conditions generally incurs a singularity, so your solution is only in $H^1$ and you shouldn't expect optimal convergence rate.

But let's assume that that switch point is in a corner of the domain and/or that you are satisfying a compatibility condition between $f, g, w$ so that there is no singularity, then to the best of my understanding, you really do get optimal convergence rate if you interpolate $g$ (no interpolation for $f,w$ is necessary -- you just need to evaluate them at quadrature points). I believe that there are papers by Willy Doerfler from the late 1980s or early to mid 1990s that show this, but it might take some searching.

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  • $\begingroup$ The existence of this paper osti.gov/biblio/… made me think that it is not so simple.. $\endgroup$
    – Leonardo
    Aug 19, 2022 at 21:03
  • $\begingroup$ But I will look into the reference you provided, many thanks $\endgroup$
    – Leonardo
    Aug 19, 2022 at 21:04
  • $\begingroup$ I had to look this particular topic up a while ago as well. I think this is the actual paper you're looking for: sciencedirect.com/science/article/pii/… $\endgroup$ Aug 19, 2022 at 21:16
  • $\begingroup$ This is close: it might be possible to adapt the arguments in the linked paper to obtain an optimal $H^1$ estimate, by employing a solution split into homogenous and inhomogeneous part. However an $L^2$ estimate is suboptimal with this method, as one would always end up with a term like $||\nabla (Eg-Eg_h)||$ ($E$ = extension) $\endgroup$
    – Leonardo
    Aug 20, 2022 at 8:38
  • $\begingroup$ With reference to the paper of Bramble, they specifically mention that some work around homogeneous Dirichlet BCs had been done (and then cite the paper of Raviart, Ciarlet), but very little around inhomogeneous ones.. on curved boundaries $\endgroup$
    – Leonardo
    Aug 20, 2022 at 8:41

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