I am trying to find a good reference on a proper weak formulation for mixed DG (Raviart Thomas and DG) formulation for a Poisson equation with mixed boundary conditions. Can anyone suggest a good reference?

Since what follows is lengthy, I am happy to offer a modicum of recompense (of your choice) for good answers.

Basically I am trying to solve is: $$ \begin{align} \mathbf{u} + \mathbf{k} \cdot \nabla p &= 0 \quad in \quad\Omega \\ \nabla \cdot \mathbf{u} &= q \quad in \quad \Omega \\ p &= g \quad on \quad \partial \Omega_D\\ \mathbf{u} \cdot \mathbf{n} &= f \quad on \quad \partial \Omega_N \end{align} $$

So I cleaned things up a bit $$ \begin{align} \mathbf{k}^{-1} \cdot \mathbf{u} + \nabla p &= 0 \quad in \quad\Omega \\ -\nabla \cdot \mathbf{u} &= -q \quad in \quad \Omega \\ p &= g \quad on \quad \partial \Omega_D\\ \mathbf{u} \cdot \mathbf{n} &= f \quad on \quad \partial \Omega_N \end{align} $$

multiplied the equations with the test functions $\mathbf{v}$ and $w$ and integrating by parts: $$ \begin{align} \int_{\Omega} (\mathbf{k}^{-1} \cdot \mathbf{u}) \mathbf{v}\mathrm{d}x + \int_{\Omega_i} \{p\} \mathbf{v} \cdot \mathbf{n} \mathrm{d}s + \int_{\Omega_D} g \mathbf{v} \cdot \mathbf{n} \mathrm{d}s + \int_{\Omega_N} p \mathbf{v} \cdot \mathbf{n} \mathrm{d}s - \int_{\Omega} p \nabla \cdot \mathbf{v} \mathrm{d}x= 0 \end{align} $$ and $$ - \int_{\Omega_D} (\mathbf{u} \cdot \mathbf{n}) w \mathrm{d}s - \int_{\Omega_N} f w \mathrm{d}s + \int_{\Omega} ( \mathbf{u} \cdot \nabla w) \mathrm{d}x = - \int_{\Omega} qw \mathrm{d}x $$

where $\{p\}=0.5(p^++p^-)$ . Since $p$ is discontinuous, I need a trace term (I think). What about $\mathbf{u}$ ? since it is a Raviart Thomas element, do I need to do anything for it? Should I change $\mathbf{u}$ to be DG (of order degree($p$)+1) as well? Does it matter?

I get the following weak formulations: $$ \begin{align} A(\{\mathbf{u},p\},\{\mathbf{v},w\}) &= (\mathbf{k}^{-1} \cdot \mathbf{u},\mathbf{v})_{\Omega} + (\{p\},\mathbf{v}\cdot \mathbf{n})_{\Omega_i}+ (p,\mathbf{v}\cdot \mathbf{n})_{\Omega_N} - (p, \nabla \cdot \mathbf{v})_{\Omega} \\ &- (\mathbf{u} \cdot \mathbf{n}, w)_{\Omega_D} + (\mathbf{u}, \nabla w)_{\Omega} \\ F(\{\mathbf{v},w \}) &= -(g,\mathbf{v} \cdot \mathbf{n})_{\Omega_D}+(f,w)_{\Omega_N}- (q,w)_{\Omega} \end{align} $$

I perform surface integrals where necessary, and all boundary conditions are imposed weakly. Would this be a stable formulation? If I do not do integration by parts for the second equation $-{\textrm{div}}\ {\mathbf u} = -q$, would imposing $\mathbf{u}\cdot \mathbf{n} = f$ on $\partial \Omega_N$ strongly be better/worse or is it irrelevant?

Thank you very much in advance.

  • 1
    $\begingroup$ You need to be careful with the normals. The term $\{p\} \mathbf{v}\cdot\mathbf{n}$ looks strange. You are by the way missing some $\partial$s in your weak form. And what is $\Omega_i$? You can derive a general form for the weak problem by integrating by parts on each element. From there the only thing that defines your method is the choice of numerical fluxes and discrete function spaces, i.e., you can either obtain mixed FEM or DG or whatever. I tried to summarize this in Ch. 4 of my thesis which you can find on my webpage (see profile). $\endgroup$ – Christian Waluga Oct 25 '14 at 16:51
  • $\begingroup$ Thank you I am looking at it right now. $\partial_i \Omega$ are the internal face boundaries where the trace/numerical-flux terms are defined. $\endgroup$ – JPL Oct 27 '14 at 17:29

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