Convergence of FEM on curved boundaries, and inhomogenous boundary data

Question

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.

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Note: cross posted on MO

Answer 1

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.