Table 1 Comparison with some HDG formulations in other papers
From: Stabilization in relation to wavenumber in HDG methods
Reference | Their notations and equations | Connection to our formulation |
|---|---|---|
Helmholtz case [2] | \({\begin{array}{l} \vec {q}_{[2]} + {\vec \nabla }{u} _{[2]} = \vec {0} \\ \mathop {\vec \nabla \cdot }\vec {q}_{[2]} - k^2 u_{[2]} = 0 \\ \hat{q}_{[2]} \cdot \vec {n} = \vec {q}_{[2]} \cdot \vec {n} + \hat{\imath }\tau _{[2]} (u _{[2]}- \hat{u} _{[2]} ) \end{array}}\) | \({ \begin{array}{ll} \tau _{[2]} & = k\, \tau \\ \hat{\imath }k u _{[2]} & = \phi \\ \vec {q} _{[2]} & =\vec {u}\end{array} }\) |
Helmholtz case [4] | \({ \begin{array}{l} \hat{\imath }k \vec {q} _{[4]} + {\vec \nabla }{u} _{[4]} = \vec {0} \\ \hat{\imath }k u _{[4]} + \mathop {\vec \nabla \cdot }\vec {q} _{[4]} = 0 \\ \hat{q} _{[4]} \cdot \vec {n} = \vec {q} _{[4]} \cdot \vec {n} + \tau _{[4]} (u _{[4]}- \hat{u} _{[4]} ) \end{array} }\) | \({ \begin{array}{ll} \tau _{[4]} & = \tau \\ u _{[4]} & = \phi \\ \vec {q} _{[4]} & =\vec {u}\end{array} }\) |
2D Maxwell case [6] | \({ \begin{array}{l} \hat{\imath }\omega _{[6]} \varepsilon _r E _{[6]} -\mathop {\nabla \times }\vec {H}_{[6]}= 0 \\ \hat{\imath }\omega _{[6]} \mu _r \vec {H}_{[6]} +\mathop {\vec \nabla \times }{E} _{[6]}= \vec {0} \\ \hat{H} _{[6]} = \vec {H}_{[6]} +\tau _{[6]} (E _{[6]} -\hat{E} _{[6]})\vec {t} \end{array} }\) | \({ \begin{array}{ll} \tau _{[6]} &= \sqrt{\frac{\varepsilon _r}{\mu _r}}\tau \\ \omega _{[6]} &= \omega \sqrt{\varepsilon _0\mu _0}\\ E _{[6]} &= \frac{1}{\sqrt{\varepsilon _r}}E,~ \vec {H}_{[6]} = \frac{1}{\sqrt{\mu _r}}\vec {H}\\ \end{array} }\) |
Maxwell case [8] | \({ \begin{array}{l} \mu \vec w _{[8]} - \mathop {\vec \nabla \times }\vec {u}_{[8]} = \vec {0} \\ \mathop {\vec \nabla \times }\vec {w}_{[8]} -\varepsilon \omega ^2 \vec {u}_{[8]} =\vec {0} \\ \hat{w} _{[8]} = \vec {w}_{[8]} + \tau _{[8]} ( \vec {u}_{[8]} - \hat{u} _{[8]} ) \times \vec {n}\end{array} }\) | \({ \begin{array}{ll} \tau _{[8]} & = \hat{\imath }\,\sqrt{ \frac{\varepsilon \omega ^2}{\mu } } \, \tau \\ \mu \vec {w}_{[8]} & =-\hat{\imath }k \vec {H},\text { with }k = \omega \sqrt{\mu \varepsilon }, \\ \vec {u}_{[8]} & =\vec {E}\end{array} }\) |