outer normal vectors <b>n</b><i><sub>i</sub></i> be labeled
accordingly. Then, on the face, we let
@f[
- \{\!\{ u \}\!\} = \frac{u_1 + u_2}2
+ \average{ u } = \frac{u_1 + u_2}2
@f]
Note, that if such an expression contains a normal vector, the
@f{multline*}
\sum_{K\in \mathbb T_h} (\nabla u, \nabla v)_K
\\
- + \sum_{F \in F_h^i} \biggl\{4\sigma_F (\{\!\{ u \mathbf n\}\!\}, \{\!\{ v \mathbf n \}\!\})_F
- - 2 (\{\!\{ \nabla u \}\!\},\{\!\{ v\mathbf n \}\!\})_F
- - 2 (\{\!\{ \nabla v \}\!\},\{\!\{ u\mathbf n \}\!\})_F
+ + \sum_{F \in F_h^i} \biggl\{4\sigma_F (\average{ u \mathbf n}, \average{ v \mathbf n })_F
+ - 2 (\average{ \nabla u },\average{ v\mathbf n })_F
+ - 2 (\average{ \nabla v },\average{ u\mathbf n })_F
\biggr\}
\\
+ \sum_{F \in F_h^b} \biggl\{2\sigma_F (u, v)_F
\eta_K^2 &= \sum_{K\in \mathbb T_h} h^2 \|f + \Delta u_h\|^2
\\
\eta_F^2 &= \sum_{F \in F_h^i} \biggl\{
- 4 \sigma_F \| \{\!\{u_h\mathbf n\}\!\} \|^2 + h \|\{\!\{\partial_n u_h\}\!\}\|^2 \biggr\}
+ 4 \sigma_F \| \average{u_h\mathbf n} \|^2 + h \|\average{\partial_n u_h}\|^2 \biggr\}
\\
\eta_B^2 &= \sum_{F \in F_h^b} 2\sigma_F \| u_h-u^D \|^2.
@f}
// them are equivalent to each other uniformly with mesh size (some not
// uniformly with polynomial degree). Here, we choose @f[ \|u\|_{1,h} =
// \sum_{K\in \mathbb T_h} \|\nabla u\|_K^2 + \sum_{F \in F_h^i}
- // 4\sigma_F\|\{\!\{ u \mathbf n\}\!\}\|^2_F + \sum_{F \in F_h^b}
+ // 4\sigma_F\|\average{ u \mathbf n}\|^2_F + \sum_{F \in F_h^b}
// 2\sigma_F\|u\|^2_F @f]
template <int dim>
is given by the following weak form
@f{align*}
&\sum_{K\in\text{cells}} \left(\nabla v_h, \nabla u_h\right)_{K}+\\
-&\sum_{F\in\text{faces}}\Big(-\left<[\![v_h]\!], \{\!\{\nabla u_h\}\!\}\right>_{F} - \left<\{\!\{\nabla v_h\}\!\}, [\![u_h]\!]\right>_{F} + \left<[\![v_h]\!], \sigma [\![u_h]\!]\right>_{F}\Big) \\
+&\sum_{F\in\text{faces}}\Big(-\left<\jump{v_h}, \average{\nabla u_h}\right>_{F} - \left<\average{\nabla v_h}, \jump{u_h}\right>_{F} + \left<\jump{v_h}, \sigma \jump{u_h}\right>_{F}\Big) \\
&= \sum_{K\in\text{cells}}\left(v_h, f\right)_{K},
@f}
-where $[\![v]\!] = v^- \mathbf{n}^- + v^+ \mathbf{n}^+ = \mathbf n^{-}
+where $\jump{v} = v^- \mathbf{n}^- + v^+ \mathbf{n}^+ = \mathbf n^{-}
\left(v^- - v^+\right)$ denotes the directed jump of the quantity $v$ from the
-two associated cells $K^-$ and $K^+$, and $\{\!\{v\}\!\}=\frac{v^- + v^+}{2}$
+two associated cells $K^-$ and $K^+$, and $\average{v}=\frac{v^- + v^+}{2}$
is the average from both sides.
The terms in the equation represent the cell integral after integration by
step-39. The penalty term is chosen such that an inverse estimate holds and
the final weak form is coercive, i.e., positive definite in the discrete
setting. The adjoint consistency term and the penalty term involve the jump
-$[\![u_h]\!]$ at the element interfaces, which disappears for the analytic
+$\jump{u_h}$ at the element interfaces, which disappears for the analytic
solution $u$. Thus, these terms are consistent with the original PDE, ensuring
that the method can retain optimal orders of convergence.
const unsigned int component = 0) const;
/**
- * Return the jump $[u]=u_{\text{cell0}} - u_{\text{cell1}}$ on the
+ * Return the jump $\jump{u}=u_{\text{cell0}} - u_{\text{cell1}}$ on the
* interface
* for the shape function @p interface_dof_index at the quadrature point
* @p q_point of component @p component.
* here uses "value here minus value there", as seen from the first cell.
*
* If this is a boundary face (at_boundary() returns true), then
- * $[u]=u_{\text{cell0}}$.
+ * $\jump{u}=u_{\text{cell0}}$.
*/
double
jump(const unsigned int interface_dof_index,
const unsigned int component = 0) const;
/**
- * Return the average $\{u\}=\frac{1}{2}u_{\text{cell0}} +
+ * Return the average $\average{u}=\frac{1}{2}u_{\text{cell0}} +
* \frac{1}{2}u_{\text{cell1}}$ on the interface
* for the shape function @p interface_dof_index at the quadrature point
* @p q_point of component @p component.
*
* If this is a boundary face (at_boundary() returns true), then
- * $\{u\}=u_{\text{cell0}}$.
+ * $\average{u}=u_{\text{cell0}}$.
*/
double
average(const unsigned int interface_dof_index,
const unsigned int component = 0) const;
/**
- * Return the average of the gradient $\{\nabla u \} = \frac{1}{2}\nabla
+ * Return the average of the gradient $\average{\nabla u} = \frac{1}{2}\nabla
* u_{\text{cell0}} + \frac{1}{2} \nabla u_{\text{cell1}}$ on the interface
* for the shape function @p interface_dof_index at the quadrature point @p
* q_point of component @p component.
*
* If this is a boundary face (at_boundary() returns true), then
- * $\{\nabla u\}=\nabla u_{\text{cell0}}$.
+ * $\average{\nabla u}=\nabla u_{\text{cell0}}$.
*/
Tensor<1, dim>
average_gradient(const unsigned int interface_dof_index,
const unsigned int component = 0) const;
/**
- * Return the average of the Hessian $\{\nabla^2 u \} = \frac{1}{2}\nabla^2
- * u_{\text{cell0}} + \frac{1}{2} \nabla^2 u_{\text{cell1}}$ on the interface
+ * Return the average of the Hessian $\average{\nabla^2 u} =
+ * \frac{1}{2}\nabla^2 u_{\text{cell0}} + \frac{1}{2} \nabla^2
+ * u_{\text{cell1}}$ on the interface
* for the shape function @p interface_dof_index at the quadrature point @p
* q_point of component @p component.
*
* If this is a boundary face (at_boundary() returns true), then
- * $\{\nabla^2 u\}=\nabla^2 u_{\text{cell0}}$.
+ * $\average{\nabla^2 u}=\nabla^2 u_{\text{cell0}}$.
*/
Tensor<2, dim>
average_hessian(const unsigned int interface_dof_index,
const unsigned int component = 0) const;
/**
- * Return the jump in the gradient $[\nabla u]=\nabla u_{\text{cell0}} -
+ * Return the jump in the gradient $\jump{\nabla u}=\nabla u_{\text{cell0}} -
* \nabla u_{\text{cell1}}$ on the interface for the shape function @p
* interface_dof_index at the quadrature point @p q_point of component @p
* component.
*
* If this is a boundary face (at_boundary() returns true), then
- * $[\nabla u]=\nabla u_{\text{cell0}}$.
+ * $\jump{\nabla u}=\nabla u_{\text{cell0}}$.
*/
Tensor<1, dim>
jump_gradient(const unsigned int interface_dof_index,
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