Variational Reconstruction
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Reconstruction from mean values with piecewise zero-mean basis:
\[
u_i(\mathbf{x})
=\overline{u}_i
+\sum_{l=1}^{N_{base}}{
u^l_i\varphi^l_i(\mathbf{x})}
\]
Functional Design:
For face \(f\) with cell on both sides \(L,R\), reconstruction functional on the face is designed as:
\[
I_f=w_g(f)\int_f{d\Gamma
\sum_{p=0}^k
{w_d(p)^2
\left\|
\mathcal{D}_pu_L
-\mathcal{D}_pu_R
\right\|_{\langle\rangle _{f,p}}^2}}
\]
where
\(w_g\) is Geometric Weight, which could well be dimensioned. An original choice is to use \(w_g=S_f^{-1}\), where \(S_f\) is area of \(f\).
\(w_d\) is Derivative Weight, which is defined to be dimensionless here.
\(\mathcal{D}_p u\) is a tensor-like (covariance only in linear transformation of space from
, no curvilinear) variable representing the derivatives of \(u\) , for
example, the Cartesian components:
$\(
[\mathcal{D}_3 u]_{ijk}
\equiv
[u]_{x_jx_jx_k}
\equiv
\frac{\partial^3u}{
\partial x_i \partial x_j\partial x_k}
\)$
\(\left\|\cdot\right\|_{\langle\rangle _{f,p}}^2\) is defined with
$\(
\left\|A\right\|_{\langle\rangle _{f,p}}^2
=\langle A,A \rangle _{f,p}
\)\(
where \)A\( is a tensor of p_th order.
The \)\langle,\rangle$ is a inner product.
Selecting Inner Product
The Normal Functional from Wang:
\[
\langle \mathcal{D}_3 u,\mathcal{D}_3 v\rangle _{f,3}
=d_f^6 \partial_{nnn}u \partial_{nnn}v
\]
X-Y aligned from Pan:
\[\begin{split}
\begin{aligned}
\langle \mathcal{D}_3 u,\mathcal{D}_3 v\rangle _{f,3}
&=(\Delta_x^3 \partial_{xxx} u)
(\Delta_x^3 \partial_{xxx} v)\\
&+(3\Delta_x^2\Delta_y \partial_{xxy} u)
(3\Delta_x^2\Delta_y \partial_{xxy} v)\\
&+(3\Delta_x\Delta_y^2 \partial_{xyy} u)
(3\Delta_x\Delta_y^2 \partial_{xyy} v)\\
&+(\Delta_y^3 \partial_{yyy} u)
(\Delta_y^3 \partial_{yyy} v)\\
\end{aligned}
\end{split}\]
Pre-Isotropic from Huang:
\[\begin{split}
\begin{aligned}
\langle \mathcal{D}_3 u,\mathcal{D}_3 v\rangle _{f,3}
&=(d_f^3 \partial_{xxx} u)
(d_f^3 \partial_{xxx} v)\\
&+(d_f^3 \partial_{xxy} u)
(d_f^3 \partial_{xxy} v)\\
&+(d_f^3 \partial_{xyy} u)
(d_f^3 \partial_{xyy} v)\\
&+(d_f^3 \partial_{yyy} u)
(d_f^3 \partial_{yyy} v)\\
\end{aligned}
\end{split}\]
Reconstruction Matrices
\[\begin{split}
\begin{aligned}
A_{mn}^i & = \sum_{j\in S_i} {
w_g(f_{i-j}) \int_{f_{i-j}}{\sum_{p=0}^k{
w_d(p)^2 \left\langle{\mathcal{D}_p(\varphi^m_i),\mathcal{D}_p(\varphi^n_i)}\right\rangle_{f_{i-j},p}
} d\Gamma}} \\
B_{mn}^{i-j} & = {
w_g(f_{i-j}) \int_{f_{i-j}}{\sum_{p=0}^k{
w_d(p)^2 \left\langle{\mathcal{D}_p(\varphi^m_i),\mathcal{D}_p(\varphi^n_j)}\right\rangle_{f_{i-j},p}
} d\Gamma}} \\
b_{m}^{i-j} & = {
w_g(f_{i-j}) \int_{f_{i-j}}{{
w_d(0)^2 \left\langle{\varphi^m_i,1}\right\rangle_{f_{i-j},p}
} d\Gamma}}
\end{aligned}
\end{split}\]
with the reconstruction system in local form:
\[
A_{mn}^i u_i^n = \sum_{j\in S_i}{
\left(B_{mn}^{i-j} u_j^n
+b_{m}^{i-j}(\overline{u}_j - \overline{u}_i)
\right)}
\]