This section is devoted to designing our SVV-stabilized triangular spectral element method for the Navier-Stokes equations. Hereafter we use boldface letters to denote vectors, vector functions, or vector spaces. Let Ω be an open, connected and bounded domain in \(\mathbb {R}^{2}\) with boundary ∂Ω assumed to be Lipschitz continuous. We use L2(Ω) to denote the space of square integrable functions in Ω. The inner product of L2(Ω) is denoted by \((u,v)_{\Omega }:=\int _ \Omega uv d\Omega \). Let H1(Ω)={v∈L2(Ω),∂xv∈L2(Ω),∂yv∈L2(Ω)}. The norm and semi-norm of H1(Ω) are denoted by ∥u∥1,Ω and |u|1,Ω respectively. Let \(H_{0}^{1}(\Omega)\) be the space of all functions in H1(Ω) having vanishing trace on ∂Ω. Let us denote the velocity vector by u, the ratio between the pressure and the (constant) density by p, and let f be a forcing known term. The Navier-Stokes equation reads:
$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{rll} D_{t} \boldsymbol{u} - \nu \Delta \boldsymbol{u} + \nabla p =& \boldsymbol{f},& \text{in} ~\Omega, \\ \nabla\cdot \boldsymbol{u} =& 0, & \text{in} ~\Omega, \end{array} \right. \end{array} $$
(1)
subject to appropriate initial and boundary conditions. In the above equations, Dtu denotes the material (Lagrangian) derivative of u with respect to time t, which can be expressed by ∂tu+u·∇u. ν is the dimensionless viscosity (the inverse of the Reynolds number).
2.1 Triangular spectral method
To clearly explain the idea, we start with a description of the spectral method in a single triangular domain △:
$$\begin{array}{@{}rcl@{}} \triangle=\left\{ (x,y):~0< x,y<1,\ 0< x+y<1 \right\}. \end{array} $$
The weak formulation of the Navier-Stokes Eq. 1 in the triangular domain △ reads: find \((\boldsymbol {u},p) \in H^{1}_{0}(\triangle)^{2} \times L_{0}^{2}(\triangle)\), such that
$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{rll} (D_{t} \boldsymbol{u},\boldsymbol{v})+ \nu (\nabla \boldsymbol{u}, \nabla \boldsymbol{v}) - (p, \nabla\cdot \boldsymbol{v}) =& (\boldsymbol{f}, \boldsymbol{v}),\ \ &\forall \boldsymbol{v}\in H^{1}_{0}(\triangle)^{2},\\ (q, \nabla\cdot \boldsymbol{u}) =& 0,\ \ & \forall q\in L_{0}^{2}(\triangle). \end{array} \right. \end{array} $$
(2)
In order to well define the triangular spectral element approximation in space to the above weak problem, we will need some notations. The one-to-one transformation between △ and the square \(\boldsymbol {\square }:=(-1,1)^{2}\) is given by the Duffy mapping x=F(ξ):
$$\begin{array}{@{}rcl@{}} x=\frac{1}{4}(1+\xi)(1-\eta),\ y=\frac{1+\eta}{2},\ \forall (\xi,\eta) \in \boldsymbol{\square}, \end{array} $$
(3)
with its inverse ξ=F−1(x) from △ to \(\boldsymbol {\square }\) by
$$\begin{array}{@{}rcl@{}} \xi=\frac{2x}{1-y}-1,\ \eta=2y-1, \ \forall (x,y)\in \triangle. \end{array} $$
(4)
(ξ,η) is often referred to as collapsed coordinate system or the Duffy coordinates. It is an easy matter to compute the Jacobian determinant, denoted by J, of the mapping F:
$$\begin{array}{@{}rcl@{}} J(\xi, \eta) = {1-\eta\over 8}. \end{array} $$
(5)
We associate a function u in △ with a function \(\widetilde {u}\) in \(\boldsymbol {\square }\) through
$$\begin{array}{@{}rcl@{}} \widetilde{u}(\xi,\eta)=u(x,y),~ x=\frac{1}{4}(1+\xi)(1-\eta), y=\frac{1+\eta}{2}, ~\forall (\xi,\eta) \in \boldsymbol{\square}. \end{array} $$
(6)
The following formulas for the gradient operators will be useful:
$$\begin{array}{@{}rcl@{}} \nabla_{\boldsymbol{x}} u = (\partial_{x} u,~\partial_{y} u)^{T}=\left(\frac{4}{1-\eta}\partial_{\xi} \widetilde{u},~~ \frac{2(1+\xi)}{1-\eta}\partial_{\xi} \widetilde{u}+2\partial_{\eta} \widetilde{u} \right)^{T}. \end{array} $$
$$\begin{array}{@{}rcl@{}} \nabla_{\boldsymbol{\xi}} \widetilde{u} = (\partial_{\xi} \widetilde{u},~\partial_{\eta} \widetilde{u})^{T}= \left(\frac{1-y}{2}\partial_{x} u, ~~\frac{x}{2(1-y)}\partial_{x}u +\frac{1}{2}\partial_{y} u\right)^{T}. \end{array} $$
The approximation space to be used consists of the rational functions generated by polynomials in the reference square through the Duffy transform. Define the rational function \(\mathcal {R}(x,y)\) in △:
$$\begin{array}{@{}rcl@{}} \mathcal {R}_{mn}(x,y)=\tilde{\mathcal {R}}_{mn}\left(\frac{2x}{1-y}-1,2y-1\right), ~\forall (x,y)\in \triangle, \end{array} $$
where \(\tilde {\mathcal {R}}_{mn}(\xi,\eta)\) be the polynomial in \(\boldsymbol {\square }\) defined by:
$$\begin{array}{@{}rcl@{}} \tilde{\mathcal {R}}_{mn}(\xi,\eta)=J_{m}^{0,0}(\xi)J_{n}^{1,0}(\eta), \ \forall (\xi,\eta)\in \boldsymbol{\square} \end{array} $$
(7)
with \(J_{k}^{\alpha,\beta }(\zeta), \zeta \in \Lambda \) being the Jacobi polynomial of degree k. Define the approximation spaces and their transformations as follows:
$$\begin{array}{@{}rcl@{}} \begin{array}{l} \mathbb{Q}_{N}(\triangle)= span\left\{\mathcal {R}_{mn}(x,y),~0\leq m,n \leq N,(x,y)\in \triangle\right\},\\ \mathbb{\widetilde{Q}}_{N}(\boldsymbol{\square})= span\{\mathcal {\widetilde{R}}_{mn}(\xi,\eta),~0\leq m,n \leq N,(\xi,\eta)\in \boldsymbol{\square}\},\\ \mathbb{Q}^{0}_{N}(\triangle)=\left\{v\in \mathbb{Q}_{N}(\triangle), v|_{\partial\triangle}=0\right\},\\ \mathbb{\widetilde{Q}}^{0}_{N}(\boldsymbol{\square})=\left\{v\in \mathbb{\widetilde{Q}}_{N}(\boldsymbol{\square}), v|_{\partial\boldsymbol{\square}}=0\right\}. \end{array} \end{array} $$
(8)
Let ξp,p=0,1,⋯,N, be the Legendre-Gauss-Lobatto points associated to LN, i.e., zeros of (1−z2)LN′(z); ωp,p=0,1,⋯,N, be the corresponding weights. We then define the discrete inner product (·,·)N on △:
$$\begin{array}{@{}rcl@{}} (\phi, \psi)_{N}:= \left(J\tilde\phi, \tilde\psi\right)_{N, \boldsymbol{\square}} := \sum_{p,q=0}^{N} \widetilde{\phi}(\xi_{p},\xi_{q})\widetilde{\psi}(\xi_{p},\xi_{q})J(\xi_{p},\xi_{q}) \omega_{p}\omega_{q}, ~\forall \phi,\psi \in C(\triangle), \end{array} $$
(9)
where J is defined in (5). Let XN and MN be the approximation spaces:
$$\begin{array}{@{}rcl@{}} \boldsymbol{X}_{N}=X_{N}^{2}, \ \ X_{N} = H^{1}_{0}(\triangle) \cap \mathbb{Q}_{N}(\triangle),\ \ M_{N}= L_{0}^{2}(\triangle) \cap \mathbb{Q}_{N-2}(\triangle). \end{array} $$
(10)
We now consider the rational spectral approximation to (2): Find uN∈XN and pN∈MN, such that
$$\begin{array}{@{}rcl@{}} \left\{ \begin{array}{rll} (D_{t} \boldsymbol{u}_{N},v_{N})_{N}+ \nu (\nabla \boldsymbol{u}_{N},\nabla \boldsymbol{v}_{N})_{N} - (p_{N}, \nabla\cdot \boldsymbol{v}_{N})_{N} =& (\boldsymbol{f}, \boldsymbol{v}_{N})_{N},& \forall \boldsymbol{v}_{N}\in \boldsymbol{X}_{N},\\ (q_{N}, \nabla\cdot \boldsymbol{u}_{N})_{N} =& 0,\ \ & \forall q_{N}\in M_{N}. \end{array} \right. \end{array} $$
(11)
It is notable that, similar to the standard spectral method for the Stokes problem, the pressure approximation space used in (11) is two degrees less than the velocity approximation space XN. This is to satisfy the well-known discrete inf-sup condition, which is necessary to avoid spurious pressure modes.
2.2 SVV stabilization
Let Λ=(−1,1),PN(Λ) is the space of polynomials of degree ≤N. We first define the spectral vanishing operator in PN(Λ), denoted by \(\mathcal {S}\), using the Legendre basis by
$$\begin{array}{@{}rcl@{}} {\mathcal S} \phi := \sum_{n=0}^{N} \hat{S}_{n} \hat{\phi}_{n} L_{n}, \quad \forall \phi\in P_{N} (\Lambda), \quad \phi = \sum_{n=0}^{N} \hat{\phi}_{n} L_{n}, \end{array} $$
where Ln is the Legendre polynomial of degree n, \(\hat {S}_{n} = 0\) if n≤mN and \(1 \geq \hat {S}_{n} \geq 0\) if n>mN. Typical choices for \(m_{N}: O (\sqrt {N})\) [23], mN=N/2 [26], or N−2 [32]. It is desirable to use a smooth variation for \(\hat {S}_{n}\) as:
$$\begin{array}{@{}rcl@{}} \hat{S}_{n} = \exp \left(- \left(\frac{N - n}{m_{N} - n}\right)^{2}\right),\, n > m_{N}. \end{array} $$
Then we define the SVV term \(-\epsilon _{N} \partial _{x} (\mathcal {S} (\partial _{x} u_{N}))\), which is written in weak form as follows:
$$\begin{array}{@{}rcl@{}} V_{N}(u_{N}, v_{N}) = \epsilon_{N} (\mathcal{S} (\partial_{x} u_{N}), \partial_{x} v_{N})_{L^{2}(\Lambda)}, \ \ \ \forall u_{N}, v_{N} \in P_{N} (\Lambda), \end{array} $$
where εN=O(1/N). Note that the SVV term may be made symmetric:
$$\begin{array}{@{}rcl@{}} V_{N}(u_{N}, v_{N}) = \epsilon_{N} \left(\mathcal{S}^{1/2} (\partial_{x} u_{N}), \mathcal{S}^{1/2} (\partial_{x} v_{N})\right)_{L^{2}(\Lambda)} \end{array} $$
with the following definition of \( \mathcal {S}^{1/2}\):
$$\begin{array}{@{}rcl@{}} \mathcal{S}^{1/2} \phi := \sum_{n=0}^{N} \sqrt{\hat{S}_{n}} \hat{\phi}_{n} L_{n}, \quad \forall \phi = \sum_{n=0}^{N} \hat{\phi}_{n} L_{n}. \end{array} $$
The SVV operator in the 2D reference domain \(\boldsymbol {\square }\) is defined in the following way. For \(\boldsymbol {u}_{N}, \boldsymbol {v}_{N} \in \mathbb {\widetilde {Q}}_{N}(\boldsymbol {\square })^{2}\), which is defined in (24), the SVV term reads
$$\begin{array}{@{}rcl@{}} V_{N}(\boldsymbol{u}_{N}, \boldsymbol{v}_{N}) = \epsilon_{N} \left(\mathcal{S}^{1/2} (\nabla \boldsymbol{u}_{N}), \mathcal{S}^{1/2} (\nabla \boldsymbol{v}_{N})\right)_{N,\boldsymbol{\square}}, \end{array} $$
where \(\mathcal {S}^{1/2}(\nabla \boldsymbol {u}_{N})\) is defined by
$$\begin{array}{@{}rcl@{}} \mathcal{S}^{1/2}{\nabla}{\boldsymbol{u}}_{N}= \left(\begin{array}{cc} \mathcal{S}^{1/2}_{\xi}(\partial_{\xi} {u}_{1,N}) & \mathcal{S}^{1/2}_{\xi} (\partial_{\xi} {u}_{2,N}) \\ \mathcal{S}^{1/2}_{\eta} (\partial_{\eta} {u}_{1,N}) & \mathcal{S}^{1/2}_{\eta}(\partial_{\eta} {u}_{2,N}) \end{array} \right) \end{array} $$
with
$$\begin{array}{@{}rcl@{}} \begin{array}{lll} \mathcal{S}^{1/2}_{\xi} \phi (\xi,\eta) := \sum_{n=0}^{N} \sqrt{\widehat{S}_{n}}\widehat{\phi}_{n}(\eta)L_{n}(\xi), & \forall \phi: \phi (\xi,\eta)= \sum_{n=0}^{N}\widehat{\phi}_{n}(\eta)L_{n}(\xi); \\ \mathcal{S}^{1/2}_{\eta} \phi (\xi,\eta) := \sum_{n=0}^{N} \sqrt{\widehat{S}_{n}}\widehat{\phi}_{n}(\xi)L_{n}(\eta), & \forall \phi : \phi (\xi,\eta)= \sum_{n=0}^{N}\widehat{\phi}_{n}(\xi)L_{n}(\eta). \end{array} \end{array} $$
Now we turn to define the SVV operator in the triangular domain △. For \(\boldsymbol {u}_{N}, \boldsymbol {v}_{N} \in \mathbb {Q}_{N}(\triangle)^{2}\), we use the Duffy mapping (3) to associate the functions \(\widetilde {\boldsymbol {u}}_{N}\) and \(\widetilde {\boldsymbol {v}}_{N}\) through (6). Doing so allows to define the SVV operator by
$$\begin{array}{@{}rcl@{}} V_{N}(\boldsymbol{u}_{N},\boldsymbol{v}_{N})=\epsilon_{N} (J{G} \mathcal{S}^{1/2}{\nabla }\tilde{\boldsymbol{u}}_{N}, {G} \mathcal{S}^{1/2} {\nabla }\tilde{\boldsymbol{v}}_{N})_{N,\boldsymbol{\square}}, \end{array} $$
(12)
where G is the Jacobian of the mapping (4):
$$\begin{array}{@{}rcl@{}} {G}= \left(\begin{array}{cc} \frac{\partial \xi}{\partial x} & \frac{\partial \eta}{\partial x} \\ \frac{\partial \xi}{\partial y} & \frac{\partial \eta}{\partial y}\\ \end{array} \right) = \left(\begin{array}{cc} \frac{4}{1-\eta} & 0 \\ {2(1+\xi)\over 1-\eta} & 2 \\ \end{array} \right). \end{array} $$
We are now in a position to propose our SVV-stabilized TSM in single domain △ as follows: find uN∈XN,pN∈MN, such that
$$\begin{array}{@{}rcl@{}} \left\{ \begin{array}{rll} (D_{t} \boldsymbol{u}_{N},\boldsymbol{v}_{N})_{N}+ \nu(\nabla \boldsymbol{u}_{N},\nabla \boldsymbol{v}_{N})_{N} \!- (p_{N}, \nabla\cdot \boldsymbol{v}_{N})_{N} +V_{N}(\boldsymbol{u}_{N},\boldsymbol{v}_{N}) =& (\boldsymbol{f}, \boldsymbol{v}_{N})_{N},& \forall \boldsymbol{v}_{N}\in \boldsymbol{X}_{N},\\ (q_{N}, \nabla\cdot \boldsymbol{u}_{N})_{N} =& 0,& \forall q_{N}\in M_{N},\\ \end{array} \right. \end{array} $$
where the stabilization term VN is defined in (12). In practice, it is highly beneficial to have the original diffusion term and the stabilization term combined together. Thus we propose to introduce the term TN(uN,vN) to replace ν(∇uN,∇vN)N+VN(uN,vN), which is defined by
$$\begin{array}{@{}rcl@{}} T_{N}(\boldsymbol{u}_{N},\boldsymbol{v}_{N})=\nu (J {G}\mathcal{T}^{1/2}{\nabla }\tilde{\boldsymbol{u}}_{N}, {G} \mathcal{T}^{1/2} {\nabla }\tilde{\boldsymbol{v}}_{N})_{N,\boldsymbol{\square}}, \end{array} $$
where
$$\begin{array}{@{}rcl@{}} \mathcal{T}^{1/2}{\nabla }\tilde{\boldsymbol{u}}_{N}= \left(\begin{array}{cc} \mathcal{T}^{1/2}_{\xi}(\partial_{\xi} \tilde{u}_{1,N}) & \mathcal{T}^{1/2}_{\xi}(\partial_{\xi} \tilde{u}_{2,N}) \\ \mathcal{T}^{1/2}_{\eta}(\partial_{\eta} \tilde{u}_{1,N}) & \mathcal{T}^{1/2}_{\eta}(\partial_{\eta} \tilde{u}_{2,N}) \end{array} \right) \end{array} $$
with
$$\begin{array}{@{}rcl@{}} \begin{array}{lll} \mathcal{T}^{1/2}_{\xi} \phi (\xi,\eta) := \sum_{n=0}^{N} \sqrt{1+\frac{\epsilon_{N}}{\nu}\widehat{S}_{n}}\widehat{\phi}_{n}(\eta)L_{n}(\xi), & \forall \phi: \phi (\xi,\eta)=\sum_{n=0}^{N}\widehat{\phi}_{n}(\eta)L_{n}(\xi); \\ \mathcal{T}^{1/2}_{\eta} \phi (\xi,\eta) := \sum_{n=0}^{N} \sqrt{1+\frac{\epsilon_{N}}{\nu}\widehat{S}_{n}}\widehat{\phi}_{n}(\xi)L_{n}(\eta), & \forall \phi: \phi (\xi,\eta)=\sum_{n=0}^{N}\widehat{\phi}_{n}(\xi)L_{n}(\eta). \end{array} \end{array} $$
Finally, the SVV-stabilized TSM for the Navier-Stokes equations reads: find uN∈XN,pN∈MN, such that
$$\begin{array}{@{}rcl@{}} \left\{ \begin{array}{rll} (D_{t} \boldsymbol{u}_{N},\boldsymbol{v}_{N})_{N}+T_{N}(\boldsymbol{u}_{N},\boldsymbol{v}_{N}) - (p_{N}, \nabla\cdot \boldsymbol{v}_{N})_{N} =& (\boldsymbol{f}, \boldsymbol{v}_{N})_{N}\ ,& \forall \boldsymbol{v}_{N}\in \boldsymbol{X}_{N},\\ (q_{N}, \nabla\cdot \boldsymbol{u}_{N})_{N} =& 0,\ \ & \forall q_{N}\in M_{N},\\ \end{array} \right. \end{array} $$
(13)
2.3 Implementation based on nodal basis
In this subsection, we give the details of the implementation of the SVV stabilization term TN(·,·). The approach described here follows what is usually done when a nodal basis is chosen.
For notation convenience, we denote by u1∈XN and v1∈XN the first component of uN and vN respectively, respectively. The first component of TN(uN,vN), denoted by \(T_{N}^{1}\), can be written as
$$\begin{array}{@{}rcl@{}} \begin{array}{lll} T_{N}^{1} &=& \nu \sum_{p,q=1}^{N}\left[G_{1} \mathcal{T}^{1/2}(\partial_{\xi} \tilde{u}_{1}) \mathcal{T}^{1/2}(\partial_{\xi} \tilde{v}_{1})+ G_{2} \mathcal{T}^{1/2}(\partial_{\eta} \tilde{u}_{1}) \mathcal{T}^{1/2}(\partial_{\eta} \tilde{v}_{1}) \right.\\ &&\left. + G_{3}(\mathcal{T}^{1/2}(\partial_{\xi} \tilde{u}_{1}) \mathcal{T}^{1/2}(\partial_{\eta} \tilde{v}_{1})+ \mathcal{T}^{1/2}(\partial_{\eta} \tilde{u}_{1}) \mathcal{T}^{1/2}(\partial_{\xi} \tilde{v}_{1})) \right](\xi_{pq})\frac{\omega_{pq}}{J(\xi_{pq})}, \end{array} \end{array} $$
where ξpq=(ξp,ηq), ωpq=ωpωq. G1,G2, and G3 are three geometric factors, defined as
$$\begin{array}{@{}rcl@{}} \begin{array}{rl} G_{1}:=& (\partial_{\eta} x)^{2}+(\partial_{\eta} y)^{2}= \frac{(1+\xi)^{2}+4}{16}, \\ G_{2}:=& (\partial_{\xi} x)^{2}+(\partial_{\xi} y)^{2}= \frac{(1-\eta)^{2}}{16}, \\ G_{3}:=& -(\partial_{\xi} x \partial_{\eta} y+\partial_{\xi} y \partial_{\eta} x) =\frac{(1+\xi)(1+\eta)}{16}. \end{array} \end{array} $$
Let hi,i=0,1,…,N be the Lagrangian polynomials associated to the Legendre-Gauss-Lobatto points {ξp,p=0,1,…,N}. Then it can be checked that the function set
$$\begin{array}{@{}rcl@{}} \left\{h_{i}(\xi)h_{j}(\eta), 0\leq i \leq M,0\leq j \leq N-1; h_{N}(\eta)\right\} \end{array} $$
forms a basis of the \(\mathbb {\widetilde {Q}}_{N}(\boldsymbol {\square })\cap H^{1}(\boldsymbol {\square })\).
Expressing u1 on this basis, i.e., \(u_{1}=\sum _{i=0}^{M}\sum _{j=0}^{N-1}u_{ij}h_{i}(\xi)h_{j}(\eta)+u_{0N}h_{N}(\eta)\), and choosing the test function v1∈XN to be each of the above basis functions, we arrive at the matrix statement of \(T_{N}^{1}\), denoted still by \(T_{N}^{1}\):
$$\begin{array}{@{}rcl@{}} \begin{array}{lll} T_{N}^{1}(m,n)&=& \nu \sum_{i=0}^{N}\frac{\rho_{in}}{|J(\xi_{in})|} G_{1,in}(D_{s})_{im} \left(\sum_{p=0}^{N}(D_{s})_{ip}u_{pn} \right) \\ && +\nu \sum_{j=0}^{N}\frac{\rho_{mj}}{|J(\xi_{mj})|} G_{2,mj}(D_{s})_{jn} \left(\sum_{q=0}^{N-1}(D_{s})_{jq}u_{mj}+(D_{s})_{jN}u_{0N} \right) \\ && +\nu \sum_{j=0}^{N-1}\frac{\rho_{mj}}{|J(\xi_{mj})|} G_{3,mj}(D_{s})_{jn} \left(\sum_{p=0}^{N}(D_{s})_{mp}u_{pj} \right) \\ && +\nu \sum_{i=0}^{N}\frac{\rho_{in}}{|J(\xi_{in})|} G_{3,in}(D_{s})_{im} \left(\sum_{q=0}^{N-1}(D_{s})_{nq}u_{iq}+(D_{s})_{nN}u_{0N} \right) \\ && \forall m=0,\ldots,N,n=0,\ldots,N-1. \end{array} \end{array} $$
$$\begin{array}{@{}rcl@{}} \begin{array}{lll} T_{N}^{1}(0,N)&=& \nu \sum_{j=0}^{N}\left(\sum_{i=0}^{N}\frac{\rho_{ij}}{|J(\xi_{ij})|} G_{2,ij}(D_{s})_{jN} \left(\sum_{q=0}^{N-1}(D_{s})_{jq}u_{iq}+(D_{s})_{jN}u_{0N} \right) \right) \\ && + \nu \sum_{j=0}^{N-1}\left (\sum_{i=0}^{N}\frac{\rho_{ij}}{|J(\xi_{ij})|} G_{3,ij}(D_{s})_{jN} \left(\sum_{p=0}^{N}(D_{s})_{ip}u_{pj} \right) \right). \end{array} \end{array} $$
Here \(D_{s}=\mathcal {T}^{1/2}D\) with D being the Legendre differentiation matrix. The matrix form of the operator \(\mathcal {T}^{1/2}\) is defined by
$$ \begin{array}{l} \mathcal{T}^{1/2}:=M^{-1}\text{diag} \left(I+\frac{\epsilon_{N}}{\nu}\widehat{S}_{n}\right)^{1/2}M, \end{array} $$
(14)
where M is the passage matrix from physical space to Legendre spectral space.
2.4 SVV stabilization in TSEM
We now briefly describe how to set up SVV-stabilized TSEM with triangle and rectangle mixed partition. Let Ω be an open bounded polygonal domain, which is decomposed as:
$$\begin{array}{@{}rcl@{}} \overline{\Omega}=\bigcup_{k=1}^{K}\overline{\Omega}_{k},~\Omega_{i} \cap \Omega_{j}= \emptyset,~i\neq j.~ \Omega_{i} ~\text{is a triangular or quadrilateral element.} \end{array} $$
Let Fk denote the mapping from the reference domain \(\boldsymbol {\square }\) to Ωk. In this case, the velocity and pressure approximation spaces are:
$$\begin{array}{@{}rcl@{}} \begin{array}{lll} X_{N}=\left\{v_{N} \in H_{0}^{1}(\Omega): v_{N}|_{\Omega_{k}}\in X_{N}^{k},1\leq k \leq K\right\},\\ M_{N}=\left\{q_{N} \in L_{0}^{2}(\Omega): q_{N} |_{\Omega_{k}}\in M_{N}^{k},1\leq k \leq K\right\}, \end{array} \end{array} $$
(15)
where
$$\begin{array}{@{}rcl@{}} \begin{array}{lll} X_{N}^{k}=\left\{v_{N}=\tilde{v}_{N}\circ F_{k}^{-1}: \tilde{v}_{N} \in \mathbb{\widetilde{Q}}_{N}(\boldsymbol{\square})\right\}, \\ [3mm] M_{N}^{k}=\left\{q_{N}=\tilde{q}_{N}\circ F_{k}^{-1}:\tilde{q}_{N} \in \mathbb{\widetilde{Q}}_{N-2}(\boldsymbol{\square})\right\}. \end{array} \end{array} $$
The SVV-stabilized TSEM in this spectral element case can be written in the same way as (13), with the SVV term taking now the element-wise sum as
$$\begin{array}{@{}rcl@{}} T_{N}(\boldsymbol{u}_{N},\boldsymbol{v}_{N})=\sum_{k=1}^{K}T_{N}^{k}(\boldsymbol{u}_{N},\boldsymbol{v}_{N}), \end{array} $$
(16)
where
$$\begin{array}{@{}rcl@{}} T_{N}^{k}(\boldsymbol{u}_{N},\boldsymbol{v}_{N})=\nu \left(J^{k} {G}^{k}\mathcal{T}^{1/2}{\nabla }\tilde{\boldsymbol{u}}_{N}, {G}^{k} \mathcal{T}^{1/2} {\nabla }\tilde{\boldsymbol{v}}_{N}\right)_{N,\boldsymbol{\square}}, k=1,2,\ldots,K, \end{array} $$
with Gk being the Jacobian of the mapping \(F^{-1}_{k}\), and Jk the Jacobian determinant of the mapping Fk.