# Convergence of FEM on curved boundaries, and inhomogenous boundary data

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

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.
• 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) Aug 20, 2022 at 8:38