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I have implemented a pretty straightforward finite element solver for the following Poisson equation. For the purposes of this question we can assume the source term and the Dirichlet data both vanish. I assume that $g\in H^{-1/2}(\partial \Omega_N) \cap L^2(\partial \Omega_N)$ (instead of just $g\in H^{-1/2}(\partial\Omega_N)$)

\begin{aligned} \mathbf{q} + \mathbf{\nabla}u &= 0 & \mathrm{in\,} \Omega,\\ \mathbf{\nabla} \cdot \mathbf{q} &= 0& \mathrm{in\ } \Omega,\\ u& = 0 &\mathrm{on\ } \partial\Omega_D,\\ \mathbf{q}\cdot \mathbf{\eta} &= g & \mathrm{on\ } \partial\Omega_N. \end{aligned}

I accounted for Neumann boundary conditions on all or some of the boundary by using Lagrange multipliers as follows.

Let the The domain $\Omega$ have a triangulation $T$ made up of simplexes $K$. We denote $$\mathbf{W}_h= \{\mathbf{p}\in \mathbf{H}^{\mathrm{div}}(\Omega)\colon \forall K\in T \quad \mathbf{p}\big|_K \in \mathrm{RT}^k(K))$$ $$V_h = \{v\in L^2(\Omega)\colon \forall K \quad v\big|_K \in P^k(K)\}$$ $$X_h = \{\mu \in L^2(\partial \Omega_N)\colon \forall K\in T\,\, \forall \mathrm{edge}\subset\partial K\cap \partial\Omega_N\quad \mu \big|_{\mathrm{edge}} \in P^k(\mathrm{edge})\}$$

Where $\mathrm{RT}^k(K)$ is the standard Raviart Thomas space and $P^k(K)$ is the space of polynomials of total degree no more than $k$.

The problem is to find the triple $(\mathbf{q}_h,u_h,\lambda_h) \in \mathbf{W}_h \times V_h \times X_h$ so that for all test functions $(\mathbf{p}, v, s) \in (W_h \times V_h \times X_h)$ we have \begin{aligned} 0 &= \int_\Omega \mathbf{p}\cdot \mathbf{q}_h\, \mathrm{d} x - \int_\Omega (\mathbf{\nabla} \cdot \mathbf{p}) u_h\, \mathrm{d} x + \int_{\partial\Omega_N} \mathbf{p} \lambda_h \cdot \mathbf{\eta}\, \mathrm{d} x \\ 0 &= \int_\Omega v (\mathbf{\nabla} \cdot \mathbf{q}_h)\, \mathrm{d} x\\ \int_{\partial\Omega_N}s g \, \mathrm{d}x&= \int_{\partial\Omega_N} s\mathbf{q}_h \cdot \mathbf{\eta}\, \mathrm{d} x \end{aligned}

The scheme works (well defined, converges to true solution, etc.) and is (I think) close to standard practice. However, in all the references I have seen in the literature (On the Mixed Finite Element Method with Lagrange Multipliers Babuska and Gatica, 2003 Numerical Methods for Partial Differential Equation and section 4.4 of A simple Introduction to the Mixed Finite Element Method by Gatica) require that the trial and test space for the Lagrange multiplier, $X_h$, have elements that are continuous across elements of the "boundary mesh".

  1. Is this scheme close to standard practice?
  2. Are there any references I could read that discuss this scheme?
  3. What problems am I allowing by choosing a piecewise discontinuous space for the Lagrange multiplier?
  4. A disadvantage of the scheme is that it requires extra regularity on $g$. Are there more disadvantages?
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    $\begingroup$ I think that you are okay having Lagrange multipliers in L2 for this case. Can I ask why you are using Lagrange multipliers? Neumann boundary for your formulation is essential. However, you can find DOFs for each edge boundary, solving the element-by-element, or rather edge-by-edge, problem, and eliminate that DOFs from the global system. At least this is what I do, and it works great. $\endgroup$ – likask Nov 16 '17 at 9:03
  • $\begingroup$ @likask I use Lagrange multipliers because I am really interested in the heat equation with time varying boundary conditions. In solving the heat equation using Lagrange multipliers we can handle the time varying boundary conditions more gracefully, as far as I can tell, than a method of eliminating constrained DOFs. $\endgroup$ – fred Nov 16 '17 at 13:57
  • $\begingroup$ I have Brezzi Boffin and Fortin on my desk :) I think that still you can do that on the edge-by-edge basis, you solve tiny problem for dofs on the face. Then you set diagonal and right hand side appropriately. You can look when I used that method for time dependent nonlinear unsaturated flow problem in soil mofem.eng.gla.ac.uk/mofem/html/mix_us_flow.html and this link to method solving for DOFs on face mofem.eng.gla.ac.uk/mofem/html/… $\endgroup$ – likask Nov 16 '17 at 14:04
  • $\begingroup$ @likask I actually would be interested in a reference that solves the above problem using the mathematically equivalent method that you use. My favorite book detailing mixed methods (Mixed Finite Element Methods and Applications, by Brezzi, Boffin, and Fortin, Springer 2013 on page 404) more or less assumes that we can write down the solution to the above problem. $\endgroup$ – fred Nov 16 '17 at 14:14
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    $\begingroup$ I have that edition. In the book is described what I am doing. Neumman bc are essential bc, so those are imposed a priori. I have practical realisation what is on pages 404/405. There is an only technical problem of determining what is tilda u, and you can see how I do it in the second link to the code. No Lagrange multipliers are needed, the method will work, but this will be not as efficient as approach suggested in the book. Feel free to take our code and modify it to your needs. $\endgroup$ – likask Nov 16 '17 at 14:39

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