Abstract
In this paper, we investigate spectral element method for fourth order problems with mixed inhomogeneous boundary conditions. Some results on the composite Legendre quasi-orthogonal approximation are established, which play important roles in spectral element method with non-uniform meshes and non-uniform approximation modes. As an example of applications, the spectral element scheme is provided for a model problem, with the convergence analysis. Numerical results demonstrate its spectral accuracy, and coincide with the analysis well. In particular, the suggested method is convenient for local mesh refinement and local mode increment, and so it works well even for the solutions changing rapidly, oscillating seriously, or behaving differently in different subdomains.
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1 Introduction
Spectral method possesses the high accuracy, and so it has been applied successfully to numerical simulations in science and engineering, see [2, 6, 12–15, 26] and the references therein. Theoretically, the larger the modes used in spectral approximations, the smaller the numerical errors. However, it is not convenient to use very large modes in actual computations. Thus, it is reasonable to study spectral element method. Some authors developed pseudospectral element method, which is also called spectral element method oftentimes, see [7, 24] and the references therein.
We considered second order problems mostly. But it is also important to deal with high order problems, see, e.g., [1, 4, 5, 11, 21, 22]. Recently, Guo, Shen and Wang [17] proposed the generalized Jacobi orthogonal approximation, which provides a useful tool for spectral method of high order problems. Meanwhile, Guo, Sun and Zhang [18] developed the generalized Jacobi quasi-orthogonal approximation, which is applicable to spectral element method for high order problems in one-dimension. Guo and Jia [16] considered the Legendre quasi-orthogonal approximation in two dimensions with its applications to spectral element method for second order problems. Whereas, so far, there is few results on the spectral element method for mixed inhomogeneous boundary value problems of high order.
The aim of this paper is to develop spectral element method for fourth order problems with mixed inhomogeneous boundary conditions. Unlike many numerical approaches using certain interpolations, the spectral element method is based on certain orthogonal approximations. As we know, the orthogonal approximations possess the high accuracy, but do not match the boundary values of approximated functions usually. Thus, the main difficulties of designing spectral element schemes for high order problems are how to match numerical solutions and their derivatives on all interfaces of adjacent elements, and how to ensure their global spectral accuracy. On the other hand, the behaviors of solutions of considered problems might be very different on different parts of domains. For raising numerical accuracy and saving computational time, it seems better to adopt non-uniform meshes and non-uniform approximation modes. It also brings certain additional difficulties. In order to remedy the above troubles, we first introduce several new Legendre quasi-orthogonal approximations defined on a square, corresponding to various kinds of boundary conditions. Then, we propose the composite Legendre quasi-orthogonal approximation on the whole complex domain with rectangle partition, and derive its error estimate precisely, which serves as the mathematical foundation of spectral element method for mixed inhomogeneous boundary value problems of fourth order. Next, we design five kinds of proper base functions corresponding to the local approximations in the different elements, on the different edges of adjacent elements and the different vertices respectively, so that the numerical solutions and their derivatives keep the continuity. The above approximation and base functions are specially appropriate for spectral element method with non-uniform meshes and non-uniform modes. Moreover, since it approximates the solutions in the interiors, at the edges and the vertices of various elements independently, with different modes under a very weak restriction, the corresponding spectral method is very convenient for local mesh refinement and local mode increment. As an example of applications, we provide the spectral element scheme for a model problem of fourth order with mixed inhomogeneous boundary conditions, and prove its global spectral accuracy. The numerical results demonstrate its high effectiveness and coincide with the analysis well. In particular, such method works well even for the solutions changing rapidly, oscillating very seriously, or behaving differently in different parts of domain.
The next section is for preliminaries. In Sect. 3, we establish the basic result on the composite Legendre quasi-orthogonal approximation. In Sect. 4, we provide the spectral element scheme for a model problem with the convergence analysis. In Sect. 5, we present some numerical results. The final section is for concluding remarks.
2 Preliminaries
Let the interval \(I=\{~x~|~a<x<b\}\) and \(h=b-a.\) The shifted Jacobi weight function
For any integer \(r\ge 0,\) we define the weighted Sobolev spaces \(H^r_{\chi ^{(\alpha ,\beta )}}(I)\) and its norm \(\Vert v\Vert _{r,\chi ^{(\alpha ,\beta )},I}\) as usual. In particular, \(H^0_{\chi ^{(\alpha ,\beta )}}(I)=L^2_{\chi ^{(\alpha ,\beta )}}(I)\) with the inner product \((u,v)_{\chi ^{(\alpha ,\beta )},I}\) and the norm \(\Vert v\Vert _{\chi ^{(\alpha ,\beta )},I}.\) We omit the subscript \(\chi ^{(\alpha ,\beta )}\) in notations whenever \(\alpha =\beta =0.\) The shifted Legendre polynomials \(L_l(\frac{1}{h}(2x-b-a))\) form a complete \(L^2(I)-\)orthogonal system.
Let \(N\) be any positive integer. \(\mathcal {P}_N(I)\) stands for the set of all algebraic polynomials of degree at most \(N.\) Throughout this paper, we denote by \(c\) a generic positive constant independent of any function and any mode \(N.\)
The shifted Legendre orthogonal projection \(P_{N,I}:L^2(I)\rightarrow \mathcal {P}_N(I)\) is defined by
According to the basic result on the standard Legendre orthogonal approximation (see page 387 of [14], as well as [19]) coupled with a affine variable transformation, we know that if \(v\in L^2(I),~\partial ^r_x v\in L^2_{\chi ^{(r,r)}}(I),\) integers \(r\ge 0\) and \(r\le N+1,\) then
Let
The orthogonal projection \(P^{1,0}_{N,I}:H^1_0(I)\rightarrow \mathcal {P}^{1,0}_N(I)\) is defined by
By virtue of Theorem 3.4 of [19], we have that if \(v\in H^1_0(I),~\partial ^r_x v\in L^2_{\chi ^{(r-1,r-1)}}(I),\) integers \(1\le r\le N+1\) and \(N\ge 1,\) then
Next, we set
and define the quasi-orthogonal projection
According to Lemma 2.4 of [20], we assert that if \(v\in H^1(I),~\partial ^r_x v\in L^2_{\chi ^{(r-1,r-1)}}(I),\) integers \(1\le r\le N+1\) and \(N\ge 1,\) then
Furthermore, let
The orthogonal projection \(P^{2,0}_{N,I}:H^2_0(I)\rightarrow \mathcal {P}^{2,0}_N(I)\) is defined by
By virtue of Theorem 2.5 of [22], we conclude that if \(v\in H^2_0(I),~\partial ^r_x v\in L^2_{\chi ^{(r-2,r-2)}}(I),\) integers \(2\le r\le N+1\) and \(N\ge 2,\) then
We now turn to the Legendre quasi-orthogonal projection in the space \(H^2(I).\) For this purpose, let \(z(x)=\frac{1}{h}(2x-b-a),\) and
It can be checked that
We introduce the auxiliary function
Furthermore, we set
Then, we define the Legendre quasi-orthogonal projection \({}_*P^2_{N,I}v\) as
Obviously,
Moveover, with the aid of Lemma 2.4 of [21], we know that if \(v\in H^2(I),~\partial ^r_x v\in L^2_{\chi ^{(r-2,r-2)}}(I),\) integers \(2\le r\le N+1\) and \(N\ge 2,\) then
In the numerical analysis of spectral method for mixed inhomogeneous boundary value problems of fourth order, we need other projections. For instance, let
We introduce the orthogonal projection \(\overline{P}^2_{N,I}:\overline{H}^2(I) \rightarrow \mathcal {\overline{P}}_N(I),\) defined by
Proposition 2.1
If \(v\in \overline{H}^2(I),~\partial ^r_x v\in L^2_{\chi ^{(r-2,r-2)}}(I),\) integers \(2\le r\le N+1\) and \(N\ge 2,\) then
Proof
Let
A direct calculation shows \(\phi (a)=\phi (b)=\partial _x\phi (b)=0.\) Thus \(\phi \in \overline{P}_N(I)\) for \(N\ge 2.\) Consequently, we use projection theorem to deduce that
Due to \(v\in \overline{H}^2(I),\) we derive that
Substituting the above inequality into (2.14), we use (2.1) to reach the result (2.13) with \(k=2.\)
We next prove the result (2.13) with \(k=0.\) Let \(g\in L^2(I)\) and consider an auxiliary problem. It is to find \(w\in \overline{H}^2(I)\) such that
Taking \(z=w\) in the above equality and using the Poincar\(\acute{e}\) inequality, we obtain \(\Vert \partial ^2_x w\Vert _I\le c\Vert g\Vert _I.\) Moreover, due to the property of elliptic equation, we have \(\partial ^4_xw(x)=g(x)\) in the sense of distributions, and so \(\Vert \partial ^4_xw\Vert _I\le c\Vert g\Vert _I.\) By taking \(z=\overline{P}^2_{N,I}v-v\) in (2.15) and using (2.13) with \(k=2,\) we verify that
Finally, we obtain from the above inequality that
which is the result (2.13) with \(k=0.\)
The result (2.13) with \(k=1\) follows from space interpolation. \(\square \)
Now, we consider the corresponding quasi-orthogonal projection. To do this, let \(z(x)\) be the same as before, and set
Let \(\overline{v}_0(x)=v(x)-\overline{v}_B(x).\) Then, we define the quasi-orthogonal projection \({}_*\overline{P}^2_{N,I}v\) as
It is easy to show that
Proposition 2.2
If \(v\in H^2(I),~\partial ^r_x v\in L^2_{\chi ^{(r-2,r-2)}}(I),\) integers \(2\le r\le N+1\) and \(N\ge 2,\) then
Proof
Clearly, \({}_*\overline{P}^2_{N,I}v(x)-v(x)=\overline{P}^2_{N,I}\overline{v}_0(x)-\overline{v}_0(x).\) Hence,
If \(r\ge 3,\) then \(\partial ^r_x\overline{v}_B(x)=0.\) In this case, the desired result (2.19) follows immediately. On the other hand, by differentiating (2.16) and using integration by parts twice, we derive that
Accordingly, we use the Cauchy inequality to obtain
Substituting the above inequality into (2.20), we reach the result (2.19) with \(r=2.\) \(\square \)
Remark 2.1
For any function \(v\in H^2(I)\) with \(v(a)=v(b)=\partial _x v(a)=0,\) we could define the Legendre quasi-orthogonal projection \({}_*{\overline{\overline{P}}}^2_{N,I}v\) and derive its error estimate similarly.
3 Legendre Quasi-Orthogonal Approximation in Two Dimensions
In this section, we develop the Legendre quasi-orthogonal approximation in two dimensions, which plays an important role in the forthcoming discussions.
3.1 Legendre Quasi-Orthogonal Approximation in Rectangles
Let \(I_i=\{x_i~|~a_i<x_i<b_i\}\) and \(h_i=b_i-a_i,~i=1,2.\) For any positive integers \(N_i,\) we define the sets \(\mathcal {P}_{N_i}(I_i)\) and \(\mathcal {P}^{2,0}_{N_i}(I_i),\) and the orthogonal projections \(P^{2,0}_{N_i,I_i}v\) from \(H^2_0(I_i)\) onto \(\mathcal {P}^{2,0}_{N_i}(I_i),\) in the same way as in Section 2. Moreover, we define the quasi-orthogonal projection \({}_*P^2_{N_i,I_i}v,\) like the definition (2.9).
Now, let \(\Omega =\{\mathbf{x}=(x_1,x_2)~|~a_i<x_i<b_i,~i=1,2\}\) with the boundary \(\partial \Omega .\) The four corners of \(\Omega \) are denoted by \(Q_1=(a_1,a_2),~Q_2=(b_1,a_2),~Q_3=(b_1,b_2)\) and \(Q_4=(a_1,b_2),\) while the four edges of \(\Omega \) are denoted by \(L_1=\overline{Q_1Q_2},~L_2=\overline{Q_2Q_3},~L_3=\overline{Q_3Q_4}\) and \(L_4=\overline{Q_4Q_1},\) respectively, see Figure 1. Let \(\chi (\mathbf {x})\) be a certain weight function. For integer \(r\ge 0,\) we define the weighted Sobolev space \(H^r_{\chi }(\Omega )\) as usual. In particular, \(H^0_{\chi }(\Omega )=L^2(\Omega )\) with the inner product \((u,v)_{\chi ,\Omega }\) and the norm \(\Vert v\Vert _{\chi ,\Omega }.\) We omit the subscript \(\chi \) whenever \(\chi (\mathbf {x})\equiv 1.\)
Rectangle \(\Omega .\)
Let \(z_i(x_i)=\frac{1}{h_i}(2x_i-b_i-a_i),\) and
Due to (2.7), we have that
Let
or equivalently,
The above function can be rewritten as the following equivalent form,
Clearly, \(w_{B,\partial \Omega }(\mathbf{x})\) is a polynomial of degree three for the variables \(x_1\) and \(x_2.\) Further, we set
By using (3.4), (3.6), (2.6), (3.2) and (3.3), we have from (3.7) that
Next, let \(N_{B,\nu }\) be positive integers, and the pair \(\mathbf{N}_B=(N_{B,1},N_{B,2},N_{B,3},N_{B,4}).\) We introduce the following projection corresponding the boundary \(\partial \Omega ,\)
It can be checked that at the four corners \(Q_\nu ,~1\le \nu \le 4,\)
Now, let the pair \(\mathbf{N}=(N_1,N_2).\) We define the Legendre quasi-orthogonal projection on the rectangle \(\Omega \) as
If \(\mathbf{N}_B=(N_1,N_2,N_1,N_2),\) then we denote the approximation \({}_*P^2_{\mathbf{N}_B,\partial \Omega }v_{B,\partial \Omega }(\mathbf{x})\) by \({}_*P^2_{\mathbf{N},\partial \Omega }v_{B,\partial \Omega }(\mathbf{x})\) for simplicity. In this case, by using (3.2) and (3.3), we obtain from (3.9) that
In order to estimate the approximation error of \({}_*P^2_{\mathbf{N},\mathbf{N}_B,\Omega }v(\mathbf{x}),\) we need some preparations.
Proposition 3.1
We have
Proof
We use the definition (2.9) to calculate \({}_*P^2_{N_2,I_2}(v-v_{B_1,I_1})(\mathbf{x}).\) Thanks to (3.2) and (3.6), the term \(v_{B,I}(x)\) in (2.9) is now replaced by \(v_{B,I_2}(\mathbf{x})-w_{B,\partial \Omega }(\mathbf{x}).\) Thus, we use (3.7) to deduce that
In other words,
Furthermore,
It can be checked that
Therefore, we use (3.12) to derive that
This ends the proof. \(\square \)
For notational convenience, let \(\mathbf{h}=(h_1,h_2)\) and \(\mathbf{r}=(r_1,r_2)\) in the forthcoming discussions. Moreover,
We shall use the following notations,
Remark 3.1
The weight functions appearing in the norms involved in the quantities \(A^{(\mu )}_{\mathbf{r},\mathbf{h},\mathbf{N},\Omega }(v)\) are helpful for covering certain singularities of approximated functions at the corners of domain. If we ignore the weights and \(r=r_1=r_2\ge 4,\) then
Proposition 3.2
If integers \(4\le r_i\le N_i+1\) for \(i=1,2,\) then
Proof
We have
with
Also, we have
with
We use (2.11) with \(k=k_1\) and \(r=2,\) and (2.11) with \(k=k_2\) and \(r=r_2-2\) successively, to derive that
Similarly,
Next, we use (2.11) with \(k=k_1\) and \(r=r_1-k_2\) to obtain
Also, thanks to (2.11) with \(k=k_2\) and \(r=r_2-k_1,\) we have
A combination of (3.15) and the previous estimates leads to
Finally, a careful calculation with (3.16) leads to the desired result (3.14).\(\square \)
Now, let \(\mathbf{r}_B=(r_{B,1},r_{B,2},r_{B,3},r_{B,4}).\) We shall use the following notations,
If \(\mathbf{N}_B=(N_1,N_2,N_1,N_2),\) then we denote the quantity \(B^{(\mu )}_{\mathbf{r}_B,\mathbf{h},\mathbf{N}_B,\partial \Omega }(v)\) by \(B^{(\mu )}_{\mathbf{r}_B,\mathbf{h},\mathbf{N},\partial \Omega }(v), \mu =0,1,2.\)
Remark 3.2
We denote by \(\partial _\tau v\) the tangential derivative on the boundary \(\partial \Omega .\) If we ignore the weights in the quantities \(B^{(\mu )}_{\mathbf{r}_B,\mathbf{h},\mathbf{N}_B,\partial \Omega }(v)\) and \(r_B=r_{B,1}=r_{B,2}=r_{B,3}=r_{B,4}\ge 3,~h=h_1=h_2,N_B=N_{B,1}=N_{B,2}=N_{B,3}=N_{B,4},\) then
Proposition 3.3
If integers \(3\le r_{B,\nu }\le N_{B,\nu }+1\) for \(1\le \nu \le 4,\) then
Proof
By virtue of (3.9), (3.7), (3.2) and (3.3), we derive that
Moreover, thanks to (3.1), a careful calculation shows that for \(i=1,2,\)
Therefore, we use (2.11) and (3.18) to reach the desired result (3.17).\(\square \)
We now in a position to estimate \( \Vert {}_*P^2_{\mathbf{N},\mathbf{N}_B,\Omega }v-v\Vert _{H^\mu (\Omega )}.\)
Theorem 3.1
If \(v\in H^{\frac{5}{ 2}+\delta }(\Omega )\) with \(\delta >0,\) integers \(4\le r_i\le N_i+1\) for \(i=1,2,\) and \(3\le r_{B,\nu }\le N_{B,\nu }+1\) for \(1\le \nu \le 4,\) then
provided that \(A^{(\mu )}_{\mathbf{r},\mathbf{h},\mathbf{N},\Omega }(v),\) \(B^{(\mu )}_{\mathbf{r}_B,\mathbf{h},\mathbf{N}_B,\partial \Omega }(v)\) and \(B^{(\mu )}_{\mathbf{r}_B,\mathbf{h},\mathbf{N},\partial \Omega }(v)\) are finite.
Proof
According to (3.11) and (3.13), we have
Like (3.17), we have
Then, a combination of (3.14), (3.17), (3.20) and (3.21) leads to the desired result (3.19).\(\square \)
Remark 3.3
As a special case, we may take \(r=r_1=r_2, r_B=r_{B,1}=r_{B,2}=r_{B,3}=r_{B,4}, h=h_1=h_2, N=N_1=N_2,\) and \( N_B=N_{B,1}=N_{B,2}=N_{B,3}=N_{B,4}.\) If we ignore the weights appearing in all norms, then (3.19) implies that for \(\mu =0,1,2,\)
Remark 3.4
If \(\mathbf{N}_B=(N_1,N_2,N_1,N_2),\) then we use (3.11) and (3.13) to find that \({}_*P^2_{\mathbf{N},\mathbf{N}_B,\Omega }v(\mathbf{x})={}_*P^2_{N_1,I_1}{}_*P^2_{N_2,I_2}v(\mathbf{x}).\) Accordingly, we see from the proof of Theorem 3.1 that
If we ignore the weights appearing in all norms, then (3.22) with \(r=r_1=r_2\) implies
But, in these cases, for keeping the continuity of \({}_*P^2_{\mathbf{N},\mathbf{N}_B,\Omega }v(\mathbf{x})\) and \(\partial _n{}_*P^2_{\mathbf{N},\mathbf{N}_B,\Omega }v(\mathbf{x})\) at interfaces of adjacent elements, we should use the uniform \(\mathbf{N}_B.\) In other words, it is only appropriate for spectral element method with uniform mode. In opposite, the definition (3.11) allows us to use different modes at different interfaces of adjacent elements, and still keep the same continuity.
3.2 Other Quasi-Orthogonal Approximations in Rectangles
For solving mixed inhomogeneous boundary value problems, we need other kinds of Legendre quasi-orthogonal approximations. For fixedness, we assume that certain Neumann or Robin boundary conditions are imposed on the edge \(L_4.\) Let \(\partial ^*\Omega =L_1\cup L_2\cup L_3,\) and
The meanings of \(\mathcal {P}_{N_i}(I_i),\) \(P^{2,0}_{N_i,I_i}v\) and \({}_*P^2_{N_i,I_i}v\) are the same as before. For any positive integers \(N_i,\) we define the sets \(\overline{\mathcal {P}}_{N_i}(I_i)\) and the orthogonal projections \(\overline{P}^2_{N_i,I_i}v\) from \(\overline{H}^2(I_i)\) onto \(\overline{\mathcal {P}}_{N_i}(I_i),\) in the same way as in Section 2. Moreover, we define the quasi-orthogonal \({}_*\overline{P}^2_{N_i,I_i}v,\) like the definition (2.17).
Let \(v_{B,I_2}(\mathbf{x})\) be the same as in (3.3), and
Furthermore,
or equivalently,
The above function can be rewritten as
Clearly, \(\overline{w}_{B,\partial \Omega }(\mathbf{x})\) is a polynomial of degree three for the variable \(x_1\) and \(x_2.\) Further, we set
By using (3.24), (3.26), (2.6), (3.23) and (3.3), we have from (3.27) that
Now, we introduce the following projection corresponding the boundary \(\partial \Omega ,\)
It can be checked that
Let the pair \(\mathbf{N}=(N_1,N_2).\) Then, we define the Legendre quasi-orthogonal projection as
If \(\mathbf{N}_B=(N_1,N_2,N_1,N_2),\) then we denote the approximation \({}_*\overline{P}^2_{\mathbf{N}_B,\partial \Omega }\overline{v}_{B,\partial \Omega }(\mathbf{x})\) by \({}_*\overline{P}^2_{\mathbf{N},\partial \Omega }\overline{v}_{B,\partial \Omega }(\mathbf{x}).\) In this case,
In order to estimate the approximation error of \({}_*\overline{P}^2_{\mathbf{N},\mathbf{N}_B,\Omega }v(\mathbf{x}),\) we need some preparations. Firstly, an argument as the proof of Proposition 3.1, with (2.9), (3.23), (3.26), (3.27) and (3.32), leads to the following result.
Proposition 3.4
We have
With the aid of (2.11) and (2.19), an argument as in the proof of Proposition 3.2 leads to the following result.
Proposition 3.5
If integers \(4\le r_i\le N_i+1\) for \(i=1,2,\) then
For notational convenience, we introduce the following quantities,
If \(\mathbf{N}_B=(N_1,N_2,N_1,N_2),\) then we denote the quantity \(\overline{B}^{(\mu )}_{\mathbf{r}_B,\mathbf{h},\mathbf{N}_B,\partial \Omega }(v)\) by \(\overline{B}^{(\mu )}_{\mathbf{r}_B,\mathbf{h},\mathbf{N},\partial \Omega }(v),~\mu =0,1,2.\)
Remark 3.5
If we ignore the weights in the quantities \(\overline{B}^{(\mu )}_{\mathbf{r}_B,\mathbf{h},\mathbf{N}_B,\partial \Omega }(v)\) and \(r_B=r_{B,1}=r_{B,2}=r_{B,3}=r_{B,4}\ge 3,~h=h_1=h_2,N_B=N_{B,1}=N_{B,2}=N_{B,3}=N_{B,4},\) then
With the aid of (3.29), (3.27), (3.23), (3.3), (2.11), (2.19) and (3.18), we follow the same line as the proof of Proposition 3.3 to derive the following result.
Proposition 3.6
If integers \(3\le r_{B,\nu }\le N_{B,\nu }+1\) for \(1\le \nu \le 4,\) then
We are now in position to estimate \( \Vert {}_*\overline{P}^2_{\mathbf{N},\mathbf{N}_B,\Omega }v-v\Vert _{H^\mu (\Omega )}.\) By virtue of (3.31), (3.33), (3.35), (3.34) and an argument similar to the proof of Theorem 3.1, we can prove the following result.
Theorem 3.2
If \(v\in H^{\frac{5}{ 2}+\delta }(\Omega )\) with \(\delta >0,\) integers \(4\le r_i\le N_i+1\) for \(i=1,2,\) and \(3\le r_{B,\nu }\le N_{B,\nu }+1\) for \(1\le \nu \le 4,\) then
provided that \(A^{(\mu )}_{\mathbf{r},\mathbf{h},\mathbf{N},\Omega }(v),\) \(\overline{B}^{(\mu )}_{\mathbf{r}_B,\mathbf{h},\mathbf{N}_B,\partial \Omega }(v)\) and \(\overline{B}^{(\mu )}_{\mathbf{r}_B,\mathbf{h},\mathbf{N},\partial \Omega }(v)\) are finite.
Remark 3.6
As a special case, we may take \(r=r_1=r_2, r_B=r_{B,1}=r_{B,2}=r_{B,3}=r_{B,4}, h=h_1=h_2, N=N_1=N_2,\) and \( N_B=N_{B,1}=N_{B,2}=N_{B,3}=N_{B,4}.\) If we ignore the weights appearing in all norms, then (3.36) implies that for \(\mu =0,1,2,\)
Remark 3.7
If \(\mathbf{N}_B=(N_1,N_2,N_1,N_2),\) then we use (3.31) and (3.33) to find that \({}_*\overline{P}^2_{\mathbf{N},\mathbf{N}_B,\Omega }v(\mathbf{x})={}_*\overline{P}^2_{N_1,I_1}{}_*P^2_{N_2,I_2}v(\mathbf{x}).\) Accordingly,
If we ignore the weights appearing in all norms, then (3.37) with \(r=r_1=r_2\) implies
Remark 3.8
If \(\partial _nv\) vanishes on other parts of the boundary, we could define the related Legendre quasi-orthogonal projection in the same way, and derive the error estimate similar to (3.36) and (3.37).
3.3 Composite Legendre Quasi-Orthogonal Approximation
We now turn to the composite Legendre quasi-orthogonal approximation on certain complex domains, which serves as the mathematical foundation of spectral element method for mixed inhomogeneous boundary value problems of fourth order.
Let \(\Omega \) be a polygon with the boundary \(\partial \Omega =\partial ^*\Omega \cup \partial ^{**}\Omega \) and \(\partial ^*\Omega \cap \partial ^{**}\Omega =\emptyset .\) We may impose Neumann or Robin boundary conditions on \(\partial ^{**}\Omega .\) In this paper, we suppose that the domain \(\Omega \) could be divided into rectangle subdomains \(\Omega _k,\) namely,
with the boundary \(\partial \Omega _k,\) the edges \(L_{k,\nu }\) and the vertices \(Q_{k,\nu }~1\le \nu \le 4.\) Besides, \(\partial ^*\Omega _k=\partial \Omega _k\cap \partial ^*\Omega \) and \(\partial ^{**}\Omega _k=\partial \Omega _k\cap \partial ^{**}\Omega .\)
We assume that the partition of \(\Omega \) satisfies the following hypotheses,
- \((\mathrm{H}_1).\) :
-
\(\overline{\Omega }=\displaystyle \cup ^M_{k=1}\overline{\Omega }_k\) and \(\Omega _{k_1}\cap \Omega _{k_2}=\emptyset \) if \(k_1\ne k_2,\)
- \((\mathrm{H}_2).\) :
-
each vertex of \(\Omega _k\) is also one of vertices of adjacent rectangles,
- \((\mathrm{H}_3).\) :
-
if \(\partial ^{**}\Omega _k\ne \emptyset ,\) then \(\Omega _k\) has at most two adjacent edges belonging to \(\partial ^{**}\Omega ,\)
- \((\mathrm{H}_4).\) :
-
if \(L_{k,\nu }\subseteq \partial ^*\Omega ,\) then \(L_{k,\nu }\nsubseteq \partial ^{**}\Omega .\)
Let \(I_{k,i}=\{x_i~|~a_{k,i}<x_i<b_{k,i}\},\) and
If \(L_{k_1,\nu _1}\) and \(L_{k_2,\nu _2}\) are the same segment, say \(L_{k_1,1}=L_{k_2,3},\) then we take \(N_{B,k_1,1}=N_{B,k_2,3}.\) The local weight function
The functions \(f^-_{k,i}(x_i),f^+_{k,i}(x_i),g^-_{k,i}(x_i)\) and \(g^+_{k,i}(x_i)\) are defined in the same way as in (3.1). We also define the local quantities \(A^{(\mu )}_{\mathbf{r}_k,\mathbf{h}_k,\mathbf{N}_k,\Omega _k}(v),\) \(B^{(\mu )}_{\mathbf{r}_{B,k},\mathbf{h}_k,\mathbf{N}_{B,k},\partial \Omega _k}(v),\) \(B^{(\mu )}_{\mathbf{r}_{B,k},\mathbf{h}_k,\mathbf{N}_k,\partial \Omega _k}(v),\) \(\overline{B}^{(\mu )}_{\mathbf{r}_{B,k},\mathbf{h}_k,\mathbf{N}_{B,k},\partial \Omega _k}(v)\) and \(\overline{B}^{(\mu )}_{\mathbf{r}_{B,k},\mathbf{h}_k,\mathbf{N}_k,\partial \Omega _k}(v)\) as in Subsections 3.1 and 3.2. Furthermore, let
If \(\mathbf{N}_{B,k}=(N_{k,1},N_{k,2},N_{k,1},N_{k,2}),\) then we denote \(\widetilde{B}^{(\mu )}_{\mathbf{r}_{B,k},\mathbf{h}_k,\mathbf{N}_{B,k},\partial \Omega _k}(v)\) by \(\widetilde{B}^{(\mu )}_{\mathbf{r}_{B,k},\mathbf{h}_k,\mathbf{N}_{k},\partial \Omega _k}(v).\)
Now, let
We define the composite Legendre quasi-orthogonal projection \({}_*P^2_{\mathbf{N},\mathbf{N}_B,\Omega }v\) on the whole domain \(\Omega \) by
where the local projections \({}_*P^2_{\mathbf{N}_k,\mathbf{N}_{B,k},\Omega _k}v\) are constructed in such a way that
- A. :
-
if \(\partial ^{**}\Omega _k=\emptyset ,\) then \({}_*P^2_{\mathbf{N}_k,\mathbf{N}_{B,k},\Omega _k}v\) is given by (3.11).
- B. :
-
if \(\partial ^{**}\Omega _k\ne \emptyset ,\) say \(\partial ^{**}\Omega _k=L_{k,1},\) then \({}_*P^2_{\mathbf{N}_k,\mathbf{N}_{B,k},\Omega _k}v\) is given by (3.31).
- C. :
-
if \(\partial ^{**}\Omega _k\ne \emptyset ,\) say \(\partial ^{**}\Omega _k=L_{k,j}\cup L_{k,j+1},\) (with \(L_{k,5}=L_{k,1}\)), then \({}_*P^2_{\mathbf{N}_k,\mathbf{N}_{B,k},\Omega _k}v\) is similar to the definition (3.31).
By using Theorems 3.1 and 3.2, we verify that if integers \(4\le r_{k,i}\le N_{k,i}+1\) and \(3\le r_{B,k}\le N_{B,k,\nu }+1\) for \(1\le k\le M,~i=1,2\) and \(1\le \nu \le 4,\) then
provided that the quantities involved at the right side of the above inequality are finite.
Remark 3.9
As a special case, we may take \(r_k=r_{k,1}=r_{k,2}, r_{B,k}=r_{B,k,1}=r_{B,k,2}=r_{B,k,3}=r_{B,k,4}, N_k=N_{k,1}=N_{k,2}, h=h_{k,1}=h_{k,2}, \) and \(N_{B}=N_{B,k,1}=N_{B,k,2}=N_{B,k,3}=N_{B,k,4},\) for \(1\le k\le M.\) If we ignore the weights appearing in all norms, then (3.39), together with Remarks 3.3 and 3.6, leads to
If \(r_k=r_{k,1}=r_{k,2}, N_{k,1}=N_{B,k,1}=N_{B,k,3}\) and \(N_{k,2}=N_{B,k,2}=N_{B,k,4}\) for \(1\le k\le M,\) then (3.39), together with Remarks 3.4 and 3.7, leads to
4 Spectral Element Method for Fourth Order Problems
In this section, we propose the spectral element method for mixed inhomogeneous boundary value problems of fourth order, which occur in many practical cases, such as laminated composite plates, free vibration of anisotropic plates under general edge conditions and so on, see [3, 10, 23, 25] and the references therein.
As in Sect. 3.3, we suppose that the domain \(\Omega \) is a union of several rectangles, with the boundary \(\partial \Omega =\partial ^*\Omega \cup \partial ^{**}\Omega ,\partial ^*\Omega \cap \partial ^{**}\Omega =\emptyset \).
Let \(d\) and \(\beta \) be nonnegative constants. We consider the following model problem,
If \(\partial ^*\Omega =\partial \Omega ,\) then the above problem is a Dirichlet boundary value problem. Otherwise, it is a mixed inhomogeneous boundary value problem. In this case, if \(\partial ^{**}\Omega =\partial \Omega \) and \(d=\beta =0,\) then we require an additional condition for ensuring the existence of solution. In fact, for any \(z,w\in H^2(\Omega ),\)
Therefore, by integrating the first equation of (4.1) and using (4.2) with \(w=\Delta U\) and \(z=1,\) we obtain
Since the considered domain \(\Omega \) is a union of several rectangles, we find that if \(\beta =0\) and \(U(\mathbf{x})=G_0(\mathbf{x})\equiv 0\) on \(\partial \Omega ,\) then the boundary condition \(\Delta U(\mathbf{x})=G_2(\mathbf{x})\) on \(\partial ^{**}\Omega ,\) implies \(\partial ^2_nU(\mathbf{x})=G_2(\mathbf{x})\) on \(\partial ^{**}\Omega .\) This is similar to the simply supported boundary condition in elastic mechanics.
We next derive the weak formulation of problem (4.1). As we know, for any \(w,z\in H^2(\Omega ),\)
Now, let
We introduce the bilinear form
By multiplying the first equation of (4.1) by \(v\in \overline{V}(\Omega )\) and integrating the resulting equality by parts, we use (4.4) to obtain the weak form of (4.1). It is to find \(U\in V(\Omega )\) such that
If \(F\in L^2(\Omega ),\) \(G_0\in H^{\frac{3}{2}}(\partial \Omega ),~G_1\in H^{\frac{1}{2}}(\partial ^{*}\Omega )\) and \(G_2\in H^{-\frac{1}{2}}(\partial ^{**}\Omega ),\) then the above problem admits a unique solution.
For solving (4.5) numerically, we first consider an auxiliary problem. We divide the domain \(\Omega \) into rectangles \(\Omega _k,~1\le k\le M.\) We also use the same notations as before, such as \(\partial \Omega _k,~\partial ^*\Omega _k,~\partial ^{**}\Omega _k,\) \(I_{k,i},h_{k,i}\) and \(L_{k,\nu }.\) Hereafter \(L_{k,\nu }\) is the \(\nu 'th\) edge of \(\partial \Omega _k\) as before. Furthermore, \({}_*P^m_{N_{B,k,\nu },I_{k,i}}v,~m=1,2,\) stand for the quasi-orthogonal projections on the edge \(L_{k,\nu }\subset \partial \Omega ,\) which are similar to the definitions (2.2) and (2.9) respectively.
We now define the following operators,
Moreover,
The auxiliary problem is to find \(W\in V^*(\Omega )\) such that
or equivalently,
We have from (4.1) and (4.7) that
According to the property of elliptic equation, we derive that
For simplicity of statements, we introduce the following quantities,
Then, with the aid of (2.3), (2.11), the inequality (4.9) implies
Remark 4.1
As a special case, we may take \(r_{B,k}=r_{B,k,1}=r_{B,k,2}=r_{B,k,3}=r_{B,k,4},h=h_{k,1}=h_{k,2},\) and \(N_{B}=N_{B,k,1}=N_{B,k,2}=N_{B,k,3}=N_{B,k,4}\) for \(1\le k\le M.\) If we ignore the weights appearing in all norms, then (4.10) implies
If \(r=r_{k,1}=r_{k,2}, N_{k,1}=N_{B,k,1}=N_{B,k,3}\) and \(N_{k,2}=N_{B,k,2}=N_{B,k,4}\) for \(1\le k\le M,\) then we have
We are going to design the spectral element scheme with non-uniform meshes and non-uniform modes. We need five kinds of base functions. To do this, let \(L_l(x)\) be the standard Legendre polynomials, and
It can be checked that
We also use the notations \(f^-_{k,i}(x_i),f^+_{k,i}(x_i),g^-_{k,i}(x_i)\) and \(g^+_{k,i}(x_i)\) as in the last section.
The first kind of base functions correspond to the subdomains \(\Omega _k,~1\le k\le M.\) If \(\partial ^{**}\Omega _k=\emptyset ,\) then we define the base functions as follows,
This function and its normal derivative vanish at the edges of \(\Omega _k.\) If \(\partial ^{**}\Omega _k=L_{k,1},\) on which we impose Neumann or Robin boundary conditions, then we define the base function
This function vanish at all edges of \(\Omega _k.\) Moreover, its normal derivative vanish at the edges \(L_{k,\nu },~\nu =2,3,4.\) If \(\partial ^{**}\Omega _k=L_{k,1}\cup L_{k,2},\) then the corresponding base function
Similarly, we can define the base functions for other kinds of \(\Omega _k.\)
We now number the edges of all subdomains as \(E_s,~1\le s\le S.\) The second kind of base functions correspond to the common edges of adjacent subdomains \(\Omega _{k_1}\) and \(\Omega _{k_2}.\) For instance, if \(E_s=L_{k_1,2}=L_{k_2,4},\) then we define the corresponding base functions as follows,
They ensure the continuity of numerical solution and its normal derivative at the common edges of adjacent subdomains.
The third kind of base functions correspond to the edges lying on the boundary \(\partial \Omega .\) For instance, if \(E_s=L_{k,1}\subset \partial ^*\Omega ,\) then the corresponding base functions are as follows,
If \(E_s=L_{k,1}\subset \partial ^{**}\Omega ,\) then the corresponding base functions are as follows,
They also ensure the continuity of numerical solution and its normal derivative at the common edges of adjacent subdomains.
We next number the vertices of all subdomains as \(V_j,~1\le j\le J.\) The fourth kind of base functions correspond to the common vertices of four adjacent subdomains \(\Omega _{k_1},\Omega _{k_2},\Omega _{k_3}\) and \(\Omega _{k_4}.\) If \(V_j=Q_{k_1,3}=Q_{k_2,4}=Q_{k_3,1}=Q_{k_4,2},\) then we define the corresponding base functions as follows,
They ensure the continuity of numerical solution and its derivatives of first order.
The fifth kind of base functions correspond to the vertices lying on \(\partial \Omega .\) For instance, if \(V_j=L_{k_1,1}\cap L_{k_2,1}\in \partial ^*\Omega ,\) then we define the corresponding base functions as follows,
Further, we can define the base functions corresponding the corners of the domain \(\Omega .\) For instance, if the corner \(V_j=Q_{k,1}\in \partial ^*\Omega ,\) then the base functions are as follows,
Now, let \(\mathcal {P}_{\mathbf{N},\mathbf{N}_B}(\Omega )\) be the finite-dimensional set spanned by the following base functions,
In addition, if \(E_s=L_{k_1,1}=L_{k_2,3},\) then \(N_s=N_{B,k_1,1}=N_{B,k_2,3},\) etc.. Furthermore,
The spectral scheme for (4.6) is to find \(w_{\mathbf{N},\mathbf{N}_B}\in V_{\mathbf{N},\mathbf{N}_B}(\Omega )\) such that
We now deal with the convergence of (4.11). To do this, we introduce the operator \(\overline{P}^2_{\mathbf{N},\mathbf{N}_B,\Omega }:V^*(\Omega )\rightarrow V_{\mathbf{N},\mathbf{N}_B}(\Omega ),\) defined by
The following two propositions play important roles in the error estimates.
Proposition 4.1
For any \(v\in V^*(\Omega )\) and \(w\in V_{\mathbf{N},\mathbf{N}_B}(\Omega ),\) we have
Proposition 4.2
If \(\Omega \) is a union of rectangles and \(v\in \overline{V}(\Omega ),\) then \(\Vert \Delta v\Vert _{\Omega }=|v|_{H^2(\Omega )}.\)
Now, we use (4.6) and (4.12) to obtain
Subtracting the above equation from (4.11) yields
This implies \(w_{\mathbf{N},\mathbf{N}_B}=\overline{P}^2_{\mathbf{N},\mathbf{N}_B,\Omega }W.\)
Clearly, \(w_{\mathbf{N},\mathbf{N}_B}(\mathbf{x})-W(\mathbf{x})=0\) on \(\partial \Omega ,\) and \(\partial _n(w_{\mathbf{N},\mathbf{N}_B}(\mathbf{x})-W(\mathbf{x}))=0\) on \(\partial ^*\Omega .\) Thus, by Proposition 4.2,
Besides, \({}_*P^2_{\mathbf{N},\mathbf{N}_B,\Omega }v\in V_{\mathbf{N},\mathbf{N}_B}(\Omega ).\) Therefore, we use (4.15), (4.13) with \(v=W\) and \(w={}_*P^2_{\mathbf{N},\mathbf{N}_B,\Omega }U,\) (3.39) with \(v=U,\) and (4.10) successively, to derive that if \(d\ge 0,\) \(4\le r_{k,i}\le N_{k,i}+1\) and \(3\le r_{B,k}\le N_{B,k,\nu }+1\) for \(1\le k\le M,~i=1,2\) and \(1\le \nu \le 4,\) then
whence
If, in addition, \(d>0,\) then
whence
Remark 4.2
If we ignore the weight functions appearing in the right sides of (4.16), and \(r_k=r_{k,1}=r_{k,2}, r_{B,k}=r_{B,k,1}=r_{B,k,2}=r_{B,k,3}=r_{B,k,4}, h=h_{k,1}=h_{k,2}, N_k=N_{k,1}=N_{k,2}, \) \(N_{B}=N_{B,k,1}=N_{B,k,2}=N_{B,k,3}=N_{B,k,4}\) for \(1\le k\le M,\) then with the help of Remarks 3.9 and 4.1, we assert that for \(d\ge 0,\)
If \(r_k=r_{k,1}=r_{k,2}, N_{k,1}=N_{B,k,1}=N_{B,k,3}\) and \(N_{k,2}=N_{B,k,2}=N_{B,k,4}\) for \(1\le k\le M,\) then we have form (4.16) that for \(d\ge 0,\)
If, in addition, \(d>0\), then the above two estimates are also valid for \(||w_{\mathbf{N},\mathbf{N}_B}-U||^2_{H^2(\Omega )} \).
Remark 4.3
In the norms involved at the right sides of the error estimates (4.16) and (4.17), there exist some weights tending to zero as the points go to the corners of elements. It is useful for covering the singularities of solutions at the corners of the boundary of considered domain.
Remark 4.4
Since we are allowed to use different mesh sizes and different approximation modes in different elements, at different common edges of adjacent elements and in different directions independently, with a weak restriction, the proposed scheme is very convenient for local mesh refinements and local mode increments.
Remark 4.5
In order to compare the above results with the corresponding results of finite element method, we take the same mesh size and the same approximation mode for all elements and ignore all weights appearing in the error estimates, then we obtain from (4.16) with trace theorem that
For fixed mode \(N,\) we have
If, in addition, \(d\ge 0,\partial ^*\Omega =\partial \Omega \) or \(d>0,\partial ^{**}\Omega \ne \emptyset ,\) then by embedding theorem,
In fact, the above error estimate with \(N=3\) and \(r=4\) is just the same as the corresponding result of high order finite element method for problem (4.1) with the rectangle \(\Omega , d=0,\) \(\partial ^*\Omega =\partial \Omega \) and \(G_0(x,y)=G_1(x,y)\equiv 0,\) given by Chen [8], which could be regarded as pseudospectral element method. We also refer to the similar results with \(r=5,\) see page 358 of [8] and Zlamal [27]. Whereas, our new method is a spectral element method and so different from the methods of [8, 27]. Moreover, the results (4.16) and (4.17) are valid for all \(4\le r\le N+1.\)
Remark 4.6
We could use an duality argument to derive the optimal estimate of \(\Vert w_{\mathbf{N},\mathbf{N}_B}-U\Vert _{L^2(\Omega )}.\) For simplicity of statements, we focus on the case with \(G_0(x,y)=G_1(x,y)\equiv 0.\) Accordingly, \(V(\Omega )=V^*(\Omega )=\overline{V}(\Omega ),V_{\mathbf{N,N}_B}(\Omega )=\overline{V}_{\mathbf{N,N}_B}(\Omega )\) and \(W=U.\) Let \(g\in L^2(\Omega ).\) We consider an auxiliary problem. It is to find \(\eta \in V(\Omega )\) such that
By taking \(z=\eta \) in (4.20), we use the Poincar\(\acute{e}\) inequality to assert that \(||\eta ||_{H^2(\Omega )}\le ||g||_\Omega \). Furthermore, by taking \(z=w_{\mathbf{N},\mathbf{N}_B}-U\) in (4.20), we obtain
It was shown in (4.14) that \(w_{\mathbf{N},\mathbf{N}_B}=\overline{P}^2_{\mathbf{N},\mathbf{N}_B,\Omega }U\in \overline{V}_{\mathbf{N,N}_B}(\Omega ).\) Thereby, we use (4.12) with \(v=U\) and \(\phi =\overline{P}^2_{\mathbf{N},\mathbf{N}_B,\Omega }\eta \) to deduce that
A combination of the above two equalities leads to
If \(r_k=r_{k,1}=r_{k,2}, N_{k,1}=N_{B,k,1}=N_{B,k,3}\) and \(N_{k,2}=N_{B,k,2}=N_{B,k,4}\) for \(1\le k\le M,\) then we have from (4.19) that
If, in addition, \(\partial \Omega =\partial ^*\Omega \) or \(d>0\), then by virtue of the property of elliptic equation, there exists \(\zeta _\Omega >0,\) such that \(\Vert \eta \Vert _{H^4(\Omega )}\le \zeta _\Omega \Vert g\Vert _{L^2(\Omega )}\) (cf. [9]). Consequently, we verify that for \(r\ge 4,\)
5 Numerical Results
In this section, we present some numerical results. In actual computations, we expand the numerical solution of (4.11) as
In addition, if the edges \(E_s\) or the vertices \(V_j\) are located on the boundary \(\partial \Omega ,\) then the corresponding coefficients \(b_{s,\mu , l}\) and \(c_{j,\mu }\) are determined by the boundary conditions \( w_{\mathbf{N},\mathbf{N}_B}(\mathbf{x})(\mathbf{x})={}_*\widetilde{P}^2_{\mathbf{N}_B,\partial \Omega }G_0(\mathbf{x})\) on \(\partial \Omega ,\) and \(\partial _n w_{\mathbf{N},\mathbf{N}_B}(\mathbf{x})={}_*\widetilde{P}^1_{\mathbf{N}_B,\partial ^*\Omega }G_1(\mathbf{x})\) on \(\partial ^*\Omega .\) By substituting (5.1) into (4.11), we obtain a system of algebra equations with the unknown coefficients \(a_{k, l,l'},~b_{s,\mu , l}\) and \(c_{j,\mu }.\)
Now, let \(\xi _{N,l}~(0\le l\le N)\) be the zeros of the Legendre polynomial \(L_{N+1}(\xi ).\) Meanwhile, \(\omega _{N,l}~(0\le l\le N)\) stand for the corresponding Christoffel numbers of the Legendre-Gauss interpolation. Furthermore, \(x^{(k,i)}_{N,l}=\frac{1}{2} (h_{k,i}\xi _{N,l}+b_{k,i}-a_{k,i}),~i=1,2.\) The errors of numerical solutions are measured by the discrete average errors
and the discrete maximum errors
In actual computation, we take the concave domain \(\Omega =\Omega _A\cup \Omega _B\) with \(\Omega _A=\{~\mathbf{x}~|~-1<x_1\le 0.5,~-1<x_2<1~\}\) and \(\Omega _B=\{~\mathbf{x}~|~0.5<x_1<1,~-0.5<x_2<0.5~\}.\)
We use (4.11) to solve problem (4.6) with \(d=\beta =0\) and the test function
which oscillates seriously for large \(\nu .\) We first use the uniform partition of domain \(\Omega ,\) with the mesh size \(h_{k,1}=h_{k,2} =h\) for \(1\le k\le M.\) In Table 1, we list the discrete errors \(E_{ave,N}\) and \(E_{max,N},\) with \(\nu =20.\) Clearly, the numerical errors decay rapidly as \(N\) increases and \(h\) decreases. This coincides the analysis well. In Table 2, we list the discrete errors \(E_{ave,N}\) and \(E_{max,N},\) with \(N=15\) and \(\nu =40.\) The spectral scheme (4.11) still works well even for the solutions oscillating seriously.
We next consider problem (4.6) with \(d=\beta =0\) and the test function
which oscillates very seriously as \(x_1\rightarrow 1.\) In this case, for raising numerical accuracy and saving work, we use non-uniform mesh sizes. More precisely, we take the mesh size \(h_{k,2}=h\) in the \(x_2\)-direction, while we take \(h_{k,1}=h\) if \(\Omega _k\subseteq \Omega _A,\) and \(h_{k,1}= h_*=\frac{1}{n} h\) if \(\Omega _k\subset \Omega _B.\) In Table 3, we list the discrete errors \(E_{max,N},\) with \(\nu =10\) and \(N=15.\) As predicted in the analysis, the spectral scheme (4.11) with non-uniform mesh sizes provides very accurate numerical results, even for the solutions oscillating seriously and locally. In fact, this is one of advantages of scheme (4.11).
Finally, we consider problem (4.6) with \(d=\beta =0\) and the test function (5.3) with very big \(\nu =100.\) In this case, the solution (5.3) oscillates extremely, and even more seriously as \(x_1\rightarrow 1.\) We take \(h_{k,1}=h\) if \(\Omega _k\subset \Omega _A,\) and \(h_{k,1}=h_*=\frac{1}{3}h\) if \(\Omega _k\subset \Omega _B.\) In Table 4, we list the discrete errors \(E_{max,N}\) with different \(h\) and \(N.\) We find that the numerical error delays very fast as \(h\) decreases and \(N\) increases. It shows again that the proposed spectral scheme (4.11) is specially appropriate for the solutions oscillating seriously. Indeed, this is another advantage of the spectral element method.
6 Concluding Remarks
In this paper, we provided the spectral element method for fourth order problems with mixed inhomogeneous boundary conditions. Since we designed the specific base functions properly, it is very appropriate for non-uniform meshes and non-uniform modes on different elements, at different common edges of adjacent elements and in different directions. Moreover, it is convenient for local mesh refinement and local mode increment. Therefore, this new approach works well for numerical simulations of problems whose solutions oscillate seriously and behave in differently ways on different parts of domain. As an example of applications, we constructed the spectral element scheme for a model problem and proved its global spectral accuracy. The numerical results demonstrated its high effectiveness. As the mathematical foundation of such new spectral element method, we proposed the composite Legendre quasi-orthogonal approximation, and established the basic approximation results. This approximation not only keeps the continuity of approximated functions and their derivatives of first order at all common edges of adjacent elements, but also possesses the global spectral accuracy on the whole complex domain.
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The work of this author is supported in part by NSF of China N.11171227, Fund for Doctoral Authority of China N.20123127110001, Fund for E-institute of Shanghai Universities N.E03004, and Leading Academic Discipline Project of Shanghai Municipal Education Commission N.J50101.
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Yu, XH., Guo, BY. Spectral Element Method for Mixed Inhomogeneous Boundary Value Problems of Fourth Order. J Sci Comput 61, 673–701 (2014). https://doi.org/10.1007/s10915-014-9844-y
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DOI: https://doi.org/10.1007/s10915-014-9844-y