must solve in an implicit system. This is because, unlike continuous finite
elements, in typical discontinuous elements there is one degree of freedom at
each vertex <i>for each of the adjacent elements</i>, rather than just one,
-and similarly for edges and faces. As an example of how fast the number of
+and similarly for edges and faces. As an example of how fast the number of
unknowns grows,
consider the <code>FE_DGP_Monomial</code> basis: each
scalar solution component is represented by polynomials of degree $p$
matrix <i>A</i> element by element (the local solution of the Dirichlet
problem) and subtract $CA^{-1}B$ from $D$. The steps in the Dirichlet-to-Neumann map concept hence correspond to
<ol>
- <li> constructing the Schur complement matrix $D-C A^{-1} B$ and right hand side $G - C A^{-1} F$ <i>locally on each cell</i>
+ <li> constructing the Schur complement matrix $D-C A^{-1} B$ and right hand side $G - C A^{-1} F$ <i>locally on each cell</i>
and inserting the contribution into the global trace matrix in the usual way,
<li> solving the Schur complement system for $\Lambda$, and
<li> solving for <i>U</i> using the second equation, given $\Lambda$.
The variable $\hat {u}$ is introduced as an additional independent variable
and is the one for which we finally set up a globally coupled linear
system. As mentioned above, it is defined on the element faces and
-discontinuous from one face to another.
+discontinuous from one face to another wherever faces meet (at
+vertices in 2d, and at edges and vertices in 3d).
Values for $u$ and $\mathbf{q}$ appearing in the numerical trace function
are taken to be the cell's interior solution restricted
-to the boundary $\partial K$.
+to the boundary $\partial K$.
The local stabilization parameter $\tau$ has effects on stability and accuracy
of HDG solutions; see the literature for a further discussion. A stabilization
@f{eqnarray*}
\tau = \frac{\kappa}{\ell} + |\mathbf{c} \cdot \mathbf{n}|
@f}
-where we set the diffusion $\kappa=1$ and the diffusion length scale to
+where we set the diffusion $\kappa=1$ and the diffusion length scale to
$\ell = \frac{1}{5}$.
The trace/skeleton variables in HDG methods are single-valued on element
- ( \nabla\cdot\mathbf{v}, u_h)_{\mathcal{T}}
+ \left<\mathbf{v}\cdot\mathbf{n}, \hat{u}_h\right>_{\partial\mathcal{T}}
&=& 0,
- \quad \forall \mathbf{v} \in \mathcal{V}_h^p,
+ \quad &&\forall \mathbf{v} \in \mathcal{V}_h^p,
\\
- (\nabla w, \mathbf{c} u_h)_{\mathcal{T}}
+ (w, \nabla \cdot \mathbf{q}_h)_{\mathcal{T}}
+ \left<w, \tau (u_h - \hat{u}_h)\right>_{\partial \mathcal{T}}
&=&
(w, f)_{\mathcal{T}},
- \quad \forall w \in \mathcal{W}_h^p,
+ \quad &&\forall w \in \mathcal{W}_h^p,
\\
\left< \mu, \hat{u}_h\mathbf{c} \cdot \mathbf{n}
+ \mathbf{q}_h\cdot \mathbf{n}
+ \tau (u_h - \hat{u}_h)\right>_{\partial \mathcal{T}}
&=&
\left<\mu, g_N\right>_{\partial\Omega_N},
- \quad \forall \mu \in \mathcal{M}_h^p.
+ \quad &&\forall \mu \in \mathcal{M}_h^p.
@f}
The unknowns $(\mathbf{q}_h, u_h)$ are referred to as local variables; they are
to denote the sum of integrals over all cells and $\left<\cdot,
\cdot\right>_{\partial \mathcal{T}} = \sum_K \left<\cdot,
\cdot\right>_{\partial K}$ to denote integration over all faces of all cells,
-i.e., interior faces are visited twice. When combining the contribution from
-both elements sharing a face, the above equation yields terms familiar for DG
-with jumps of the solution over the cell boundaries.
+i.e., interior faces are visited twice, once from each side and with
+the corresponding normal vectors. When combining the contribution from
+both elements sharing a face, the above equation yields terms familiar
+from the DG method, with jumps of the solution over the cell boundaries.
In the equation above, the space $\mathcal {W}_h^{p}$ for the scalar variable
<i>u<sub>h</sub></i> is defined as the space of functions that are tensor
<ol>
<li> The matrix $A$ consists of local-local coupling terms. These arise when the
local weighting functions $(\mathbf{v}, w)$ multiply the local solution terms
- $(\mathbf{q}_h, u_h)$.
+ $(\mathbf{q}_h, u_h)$. Because the elements are discontinuous, $A$
+ is block diagonal.
<li> The matrix $B$ represents the local-face coupling. These are the terms
with weighting functions $(\mathbf{v}, w)$ multiplying the skeleton variable
$\hat{u}_h$.
</ol>
We now introduce a new variable $u_h^* \in \mathcal{V}_h^{p+1}$, which we find
-by the expression $|\kappa \nabla u_h^* + \mathbf{q}_h|^2$ over the cell
+by minimizing the expression $|\kappa \nabla u_h^* + \mathbf{q}_h|^2$ over the cell
<i>K</i> under the constraint $\left(1, u_h^*\right)_K &=& \left(1,
-u_h\right)_K$. This translates to the following system of equations:
+u_h\right)_K$. The constraint is necessary because the minimization
+functional does not determine the constant part of $u_h^*$. This
+translates to the following system of equations:
@f{eqnarray*}
\left(1, u_h^*\right)_K &=& \left(1, u_h\right)_K\\
-\left(\nabla w_h^*, \kappa \nabla u_h^*\right)_K &=&
+\left(\nabla w_h^*, \kappa \nabla u_h^*\right)_K &=&
-\left(\nabla w_h^*, \mathbf{q}_h\right)_K
\quad \text{for all } w_h^* \in \mathcal Q^{p+1}.
@f}
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