## 2 Spectral/hp Element Methods
### Spectral/hp Element Methods  
A detailed content of the spectral/hp element methods can refer to [PERSON] et al. and [PERSON] et al. [25; 26].  
Generally, for the partial differential equations (PDEs), \(Lu=0\), which is defined on a domain \(\Omega\), the domain \(\Omega\) is divided into n-dimensional finite element meshes \(\Omega_{\epsilon_{1}}\), \(\Omega_{\epsilon_{2}}\),\(\cdots\cdots\), \(\Omega_{\epsilon_{N}}\), \(n=1\), 2, 3, \(\Omega=\underset{i=1}{\cup}\Omega_{\epsilon_{i}}\), \(\Omega_{\epsilon_{i}}\cap\Omega_{\epsilon_{j}}=\partial\Omega_{\epsilon_{ \epsilon_{j}}}\). We seek the weak solution of the PDEs \(Lu=0\), \(u\Big{|}_{\Omega_{\epsilon_{i}}}\in W^{1,2}(\Omega_{\epsilon_{i}})\) (Sobolev space). We find \(u\in H^{1}(\Omega)\) such that \(a(u,v)=f(v)\), \(\forall v\in H^{1}(\Omega)\), where \(a(\cdot,\ \cdot)\) is a symmetric bilinear form, \(f(\cdot)\) is a linear form, and \(H^{1}(\Omega)\) is a Sobolev space, which is defined as follows:  
\[H^{1}(\Omega)=W^{1,2}(\Omega)=\Big{\{}u\in L^{2}(\Omega)\Big{|}D^{\ast}u\in L^ {2}(\Omega),\ \forall\Big{|}\alpha\Big{|}\leq 1\Big{\}}. \tag{3}\]  
In a finite dimensional subspace \(S_{N}\subset H^{1}(\Omega)\), we seek \(u^{\ast}\in S_{N}\) such that \(a(u^{\ast},v^{\ast})=f(v^{\ast})\), \(\forall v^{\ast}\in S_{N}\); in order to keep continuity across elements, \(S_{N}\subset C^{0}\) is required. Suppose \(u^{\ast}\) can be written as a linear combination of trial functions \(\Phi_{i}(x)\), i.e.,\(u^{\ast}(x)=\sum\limits_{i=1}^{N}C_{i}\Phi_{i}(x)\), where \(\Phi_{i}(x)\) is defined on \(\Omega\). Then, the problem converts to determining the coefficients \(C_{i}\). Next, the residual \(R=Lu^{\ast}\) are restricted so that its \(L^{2}\) inner product is zero with respect to the test functions \(\Psi_{i}(x)\). As it is well known to all, for the Galerkin projections, we let \(\Psi_{i}(x)=\Phi_{i}(x)\).  
For the construction of the global basis functions \(\Phi_{i}(x)\), the contributions from every element \(\Omega_{e_{i}}\) in the solution domain should be considered. We mapped every \(\Omega_{e_{i}}\) into a standard element \(\Omega_{\text{sf}}\subset[-1,\ \ 1]^{n}(n=1,\,2,\,3)\) by an isoparametric mapping. Similarly, the mappings of high-order elements can refer to [PERSON] et al. [26].  
We denote the space \(\varphi^{p}\) as the polynomials space of order \(p\) defined on the standard element \(\Omega_{\text{sf}}\). In one-dimensional case, the local polynomial basis functions on each standard element \(\Omega_{\text{st}}\) can be constructed by the following hierarchical basis functions [25]:  
\[N_{1}(\xi)=\frac{1-\xi}{2},\ \ N_{2}(\xi)=\frac{1+\xi}{2},\ N_{i}(\xi)=\phi_{i-1 }(\xi),\ \ \ i=3,\,4,\cdots,\ p+1,\]  
where \(\phi_{j}\) is defined by the following Legendre polynomial \(P_{j-1}\),  
\[\phi_{j}(\xi)=\sqrt{\frac{2j-1}{2}}\int_{-1}^{\xi}P_{j-1}(t)dt,\qquad j=2,\,3,\ldots.\]  
In two- and three-dimensional cases, the tensorial basis functions are used, which are derived as the tensor-product of the one-dimensional basis functions on one-dimensional, two-dimensional or three-dimensional elements. If we define the local polynomial basis functions by the Lagrange interpolation polynomials, then it will lead to the spectral element method.