decomposing the domain \(\Omega\) into \(N_e\) conforming
non-overlapping triangular elements \(\Omega_e\).
\[\begin{equation}
\Omega = \bigcup_{e = 1}^{N_e} \Omega_e
\end{equation}\]
nonsingular mapping \(x = \Psi(\mathbf{\xi})\) which defines a transformation from the physical Cartesian coordinate system to the local reference coordinate system defined on the reference triangle.
local elementwise solution \(\mathbf{q}\) by an N th order polynomial in \(\mathbf{\xi}\) as
\[\begin{equation}
\mathbf{q}_N (\mathbf{\xi}) = \sum_{i = 1}^{M_N} \psi_i (\mathbf{\xi}) \mathbf{q}_N (\mathbf{\xi}_i)
\end{equation}\]
where \(\mathbf{\xi}_i\) represents \(M = \frac{1}{2} ( N + 1)( N + 2)\) interpolation points and \(\psi_i (\mathbf{\xi})\) are the associatedmultivariate Lagrange polynomials.
an explicit formula for the Lagrange basis —— reference to an easily constructed orthonormal PKD polynomial basis and the generalized Vandermonde matrix.
通过正交多项式和Vandermonde构造参考单元上Lagrange基函数。
\(\int_{\Omega_e} f(x) g(x) dx = \sum_{i = 1}^{M_C} \omega_i^e \left| J^e(\mathbf{\xi}_i) \right| f(\mathbf{\xi}_i) g(\mathbf{\xi}_i)\)
where \(M_C\) is a function of \(C\) which represents the order of the cubature approximation.
\(\int_{\Gamma_e} f(x) g(x) dx = \sum_{i = 0}^{Q} \omega_i^s \left| J^s(\mathbf{\xi}_i) \right| f(\mathbf{\xi}_i) g(\mathbf{\xi}_i)\)
where \(Q\) represents the order of the quadrature approximation. Using the Gauss quadrature, we
can use \(Q = N\) to achieve order \(2N\) accuracy.
将方程左乘质量矩阵的逆并除以雅克比系数,可得
\[\begin{equation}
\frac{\partial \mathbf{q}^e_i}{\partial t} + \left( \hat{D}_{ij}^{\xi} \xi_x^e + \hat{D}_{ij}^{\eta} \eta_x^e \right) \mathbf{f}_j^e + \left( \hat{D}_{ij}^{\xi} \xi_y^e + \hat{D}_{ij}^{\eta} \eta_y^e \right) \mathbf{g}_j^e - S_i^e = \frac{\left| J^s \right|}{\left| J^e \right|} \hat{M}_{ij}^s \left[ n_x^s \left( \mathbf{f}^e - \mathbf{f}^* \right)_j + n_y^s \left( \mathbf{g}^e - \mathbf{g}^* \right)_j \right]
\end{equation}\]
where the matrices are defined as
\[\begin{equation}
\begin{array}{lll}
\hat{D}_{ij}^{\xi} = M_{ik}^{-1} D_{kj}^{\xi}, & \hat{D}_{ij}^{\eta} = M_{ik}^{-1} D_{kj}^{\eta}, &
\hat{M}_{ij}^{s} = M_{ik}^{-1} M_{kj}^{\xi},
\end{array}
\end{equation}\]
where
\[\begin{equation}
\begin{array}{ll}
M_{ij} = \sum_{k = 1}^{M_C} \omega_k \psi_{ik} \phi_{jk}, & M_{ij}^s = \sum_{k = 1}^{M_Q} \omega_k \psi_{ik} \phi_{jk} \cr
D_{ij}^{\xi} = \sum_{k = 1}^{M_C} \omega_k \psi_{ik} \frac{\partial \phi_{jk}}{\partial \xi}, & D_{ij}^{\eta} = \sum_{k = 1}^{M_C} \omega_k \psi_{ik} \frac{\partial \phi_{jk}}{\partial \eta}
\end{array}
\end{equation}\]
\(M_C\) and \(M_Q\) denote the number of cubature (two dimensional) and quadrature (one dimensional) integration points required to achieve order 2N accuracy, and \(\psi_{ik}\) represents the function \(\psi\) at the \(i=1, \cdots,M_N\) interpolation points evaluated at the integration point k.
Since the mass matrix is constant (i.e. not a function of x) then, using Equations above, we can move the mass matrix inside the summations which are the discrete representations of the continuous integrals. This then gives
\[\begin{equation}
\begin{array}{ll}
\hat{M}_{ij}^{s} = \sum_{k = 1}^{M_Q} \omega_k \hat{\psi}_{ik} \psi_{jk}, & \hat{D}_{ij}^{\xi} = \sum_{k = 1}^{M_C} \omega_k \hat{\psi}_{ik} \frac{\partial \psi_{jk}}{\partial \xi}, & \hat{D}_{ij}^{\eta} = \sum_{k = 1}^{M_C} \omega_k \hat{\psi}_{ik} \frac{\partial \psi_{jk}}{\partial \eta}
\end{array}
\end{equation}\]
where
\[\begin{equation}
\hat{\psi}_i = M_{ik}^{-1} \psi_k
\end{equation}\]
根据
\(D_{ij}^{\xi} = \sum_{k = 1}^{M_C} \omega_k \psi_{ik} \frac{\partial \psi_{jk}}{\partial \xi}\)
我们可以将 \(D_{ij}^{\xi}\) 写为如下矩阵相乘形式
\[\begin{equation}
D_{ij}^{\xi} = \begin{bmatrix}
\omega_1 \psi_{11}, \omega_2 \psi_{12}, \cdots, \omega_{M_C} \psi_{1{M_C}}
\end{bmatrix}
\begin{bmatrix}
\frac{\partial \psi_{11}}{\partial \xi} \cr \frac{\partial \psi_{12}}{\partial \xi} \cr
\cdots \cr
\frac{\partial \psi_{1{M_C}}}{\partial \xi}
\end{bmatrix}
\end{equation}\]
因此
\[D^{\xi} = \begin{bmatrix}
\omega_1 \psi_{11}, \omega_2 \psi_{12}, \cdots, \omega_{M_C} \psi_{1{M_C}} \cr
\omega_1 \psi_{21}, \omega_2 \psi_{22}, \cdots, \omega_{M_C} \psi_{2{M_C}} \cr
\cdots \cr
\omega_1 \psi_{{M_C}1}, \omega_2 \psi_{{M_C}2}, \cdots, \omega_{M_C} \psi_{{M_C}{M_C}} \cr
\end{bmatrix}
\begin{bmatrix}
\frac{\partial \psi_{11}}{\partial \xi}, & \frac{\partial \psi_{21}}{\partial \xi}, & \cdots & \frac{\partial \psi_{{M_C}1}}{\partial \xi} \cr \frac{\partial \psi_{12}}{\partial \xi}, & \frac{\partial \psi_{22}}{\partial \xi}, & \cdots & \frac{\partial \psi_{{M_C}2}}{\partial \xi} \cr
\cdots \cr
\frac{\partial \psi_{1{M_C}}}{\partial \xi}, & \frac{\partial \psi_{2{M_C}}}{\partial \xi}, & \cdots & \frac{\partial \psi_{{M_C}{M_C}}}{\partial \xi}
\end{bmatrix}\]
因此
\[\hat{D}^{\xi} = M^{-1} \begin{bmatrix}
\omega_1 \psi_{11}, \omega_2 \psi_{12}, \cdots, \omega_{M_C} \psi_{1{M_C}} \cr
\omega_1 \psi_{21}, \omega_2 \psi_{22}, \cdots, \omega_{M_C} \psi_{2{M_C}} \cr
\cdots \cr
\omega_1 \psi_{{M_C}1}, \omega_2 \psi_{{M_C}2}, \cdots, \omega_{M_C} \psi_{{M_C}{M_C}} \cr
\end{bmatrix}
\begin{bmatrix}
\frac{\partial \psi_{11}}{\partial \xi}, & \frac{\partial \psi_{21}}{\partial \xi}, & \cdots & \frac{\partial \psi_{{M_C}1}}{\partial \xi} \cr \frac{\partial \psi_{12}}{\partial \xi}, & \frac{\partial \psi_{22}}{\partial \xi}, & \cdots & \frac{\partial \psi_{{M_C}2}}{\partial \xi} \cr
\cdots \cr
\frac{\partial \psi_{1{M_C}}}{\partial \xi}, & \frac{\partial \psi_{2{M_C}}}{\partial \xi}, & \cdots & \frac{\partial \psi_{{M_C}{M_C}}}{\partial \xi}
\end{bmatrix}\]
[1]: GIRALDO F X, WARBURTON T. A high-order triangular discontinuous Galerkin oceanic shallow water model[J]. International Journal for Numerical Methods in Fluids, 2008, 56: 899–925.
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