Complementary results for the spectral analysis of matrices in Galerkin methods with GB-splines

TL;DR

This paper presents new spectral analysis results for matrices arising in Galerkin methods using highly smooth generalized B-splines, enhancing understanding of their spectral properties.

Contribution

It provides novel spectral analysis results specifically for matrices in Galerkin methods with high-smoothness GB-splines, which were not previously documented.

Findings

New spectral bounds for matrices in Galerkin GB-spline methods

Enhanced understanding of eigenvalue distributions

Implications for numerical stability and efficiency

Abstract

We collect some new results relative to the study of the spectral analysis of matrices in Galerkin methods based on generalized B-splines with high smoothness.

Equations

$$C_p\frac{\Vert \mathbf{x}{\Vert }^2}{n}\le {\Vert \sum _{i=1}^{n+p-2}{x}_i{N}_{i+1,p}\Vert }_{L_2([0,1])}^2\le {\overline{C}}_p\frac{\Vert \mathbf{x}{\Vert }^2}{n},$$

$$K_p^{{\mathbb{Q}}_{\alpha }}$$

$$K_p^{{\mathbb{Q}}_{[0,\alpha ]}}$$

$$K_p^{{\mathbb{H}}_{[0,\alpha ]}}$$

$$K_p^{{\mathbb{T}}_{[0,\alpha ]}}$$

$${\overline{K}}_p^{{\mathbb{H}}_{[0,\alpha ]}}$$

$${\overline{K}}_p^{{\mathbb{T}}_{[0,\alpha ]}}$$

$${\Vert \sum _{i=1}^{n+p-2}{x}_i{K}_p^{{\mathbb{Q}}_{\alpha }}{N}_{i+1,p}\Vert }^2$$

$${\Vert \sum _{i=1}^{n+p-2}{x}_i{K}_p^{{\mathbb{Q}}_{[0,\alpha ]}}{N}_{i+1,p}\Vert }^2$$

$$C_p^{{\mathbb{Q}}_{\alpha }}\frac{\Vert \mathbf{x}{\Vert }^2}{n}$$

$$C_p^{{\mathbb{Q}}_{[0,\alpha ]}}\frac{\Vert \mathbf{x}{\Vert }^2}{n}$$

$$\Vert v{\Vert }_{L_2([0,1])}\le \frac{1}{\pi }\Vert v^{\prime }{\Vert }_{L_2([0,1])},\forall v\in H_0^1([0,1]),$$

$$\underset{n\to \infty }{\lim }(n^2{\lambda }_j(K_{n,p}^{{\mathbb{Q}}_{\mu }}))=j^2{\pi }^2.$$

$$|{\lambda }_{\min }(A_{n,p}^{{\mathbb{Q}}_{\mu }})|\ge {\lambda }_{\min }(\text{Re}{A}_{n,p}^{{\mathbb{Q}}_{\mu }})\ge \frac{C_p^{{\mathbb{Q}}_{\nu }}({\pi }^2+\gamma )}{n},$$

$$\text{Re}X:=\frac{X+X^{\ast }}{2},\text{Im}X:=\frac{X-X^{\ast }}{2\text{i}}.$$

$$A_{n,p}^{{\mathbb{Q}}_{\mu }}=n{K}_{n,p}^{{\mathbb{Q}}_{\mu }}+\beta H_{n,p}^{{\mathbb{Q}}_{\mu }}+\frac{\gamma }{n}{M}_{n,p}^{{\mathbb{Q}}_{\mu }},$$

$$\text{Re}{A}_{n,p}^{{\mathbb{Q}}_{\mu }}=n{K}_{n,p}^{{\mathbb{Q}}_{\mu }}+\frac{\gamma }{n}{M}_{n,p}^{{\mathbb{Q}}_{\mu }}.$$

$${\lambda }_{\min }(\text{Re}{A}_{n,p}^{{\mathbb{Q}}_{\mu }})\ge {\lambda }_{\min }(n{K}_{n,p}^{{\mathbb{Q}}_{\mu }})+{\lambda }_{\min }\left(\frac{\gamma }{n}{M}_{n,p}^{{\mathbb{Q}}_{\mu }}\right)\ge n\frac{{\pi }^2{C}_p^{{\mathbb{Q}}_{\nu }}}{n^2}+\frac{\gamma }{n}{C}_p^{{\mathbb{Q}}_{\nu }}=\frac{C_p^{{\mathbb{Q}}_{\nu }}({\pi }^2+\gamma )}{n}.$$

$$\sigma (X)\subseteq [{\lambda }_{\min }(\text{Re}X),{\lambda }_{\max }(\text{Re}X)]\times [{\lambda }_{\min }(\text{Im}X),{\lambda }_{\max }(\text{Im}X)]\subset \mathbb{C},\forall X\in {\mathbb{C}}^{m\times m}.$$

$${\kappa }_2(A_{n,p}^{{\mathbb{Q}}_{\mu }}):=\Vert A_{n,p}^{{\mathbb{Q}}_{\mu }}\Vert \text{mbox}\Vert (A_{n,p}^{{\mathbb{Q}}_{\mu }}{)}^{-1}\Vert .$$

$$\Vert X\Vert =s_1(X)\ge s_2(X)\ge \cdots \ge s_m(X),{\lambda }_1(\text{Re}X)\ge {\lambda }_2(\text{Re}X)\ge \cdots {\lambda }_m(\text{Re}X)$$

$$s_j(X)\ge {\lambda }_j(\text{Re}X),\forall j=1,\dots ,m.$$

$${\kappa }_2(A_{n,p}^{{\mathbb{Q}}_{\mu }})\le {\alpha }_p{n}^2,\forall n\ge 2.$$

$$\Vert A_{n,p}^{{\mathbb{Q}}_{\mu }}\Vert$$

$$\Vert (A_{n,p}^{{\mathbb{Q}}_{\mu }}{)}^{-1}\Vert =\frac{1}{s_{n+p-2}(A_{n,p}^{{\mathbb{Q}}_{\mu }})}\le \frac{1}{{\lambda }_{\min }(\text{Re}{A}_{n,p}^{{\mathbb{Q}}_{\mu }})}\le \frac{n}{C_p^{{\mathbb{Q}}_{\nu }}({\pi }^2+\gamma )}.$$

$${\kappa }_2(A_{n,p}^{{\mathbb{Q}}_{\mu }})\le \frac{C_{p,\alpha }{n}^2+2n|\beta |+\gamma (p+1)}{C_p^{{\mathbb{Q}}_{\nu }}({\pi }^2+\gamma )}\le \frac{C_{p,\alpha }{n}^2+n^2|\beta |+\gamma (p+1)(n^2/4)}{C_p^{{\mathbb{Q}}_{\nu }}({\pi }^2+\gamma )}.$$

$${\lambda }_j(T_m(f))-m_f{\sim }_{m\to \infty }{c}_{s,j}\frac{f^{(2s)}({\theta }_{\min })}{(2s)!}\frac{1}{m^{2s}},$$

$$\{\begin{array}{rl}& (-1{)}^s{u}^{(2s)}(x)=f(x),\text{mbox}for0<x<1,\\ & u(0)=u^{\prime }(0)=\cdots =u^{(s-1)}(0)=0,u(1)=u^{\prime }(1)=\cdots =u^{(s-1)}(1)=0,\end{array}$$

$$T_{m_1,m_2}(g):=\left[\begin{array}{ccccc}{G}_0& G_{-1}& \cdots & \cdots & G_{-(m_1-1)}\\ G_1& \ddots & \ddots & & ⋮\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ ⋮& & \ddots & \ddots & G_{-1}\\ G_{m_1-1}& \cdots & \cdots & G_1& G_0\end{array}\right]\in {\mathbb{C}}^{m_1{m}_2\times m_1{m}_2},$$

$$G_k:=\left[\begin{array}{ccccc}{g}_{k,0}& g_{k,-1}& \cdots & \cdots & g_{k,-(m_2-1)}\\ g_{k,1}& \ddots & \ddots & & ⋮\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ ⋮& & \ddots & \ddots & g_{k,-1}\\ g_{k,m_2-1}& \cdots & \cdots & g_{k,1}& g_{k,0}\end{array}\right]\in {\mathbb{C}}^{m_2\times m_2},$$

$$g_{k,l}:=\frac{1}{(2\pi {)}^2}{\int }_{-\pi }^{\pi }{\int }_{-\pi }^{\pi }g({\theta }_1,{\theta }_2){\text{e}}^{-\text{i}(k{\theta }_1+l{\theta }_2)}d{\theta }_{1}d{\theta }_2$$

$$f\otimes h:[-\pi ,\pi {]}^2\to \mathbb{R},(f\otimes h)({\theta }_1,{\theta }_2):=f({\theta }_1)h({\theta }_2).$$

$$T_{m_1}(f)\otimes T_{m_2}(h)=T_{m_1,m_2}(f\otimes h).$$

Full text

Complementary results for the spectral analysis of matrices in Galerkin methods with GB-splines

Fabio Roman111University of Torino, Dipartimento di Matematica, Via Carlo Alberto 10, 10123 Torino. Tel: +39 011 6702827; Fax: +39 011 6702878; e-mail: fabio.roman@unito.it

1 Introduction

In this short note we collect some results relative to the study of the spectral analysis of matrices in Galerkin methods based on generalized B-splines with high smoothness, depicted in [1].

They are the generalization of the ones reported in [2] for the polynomial case, which were studied also in the generalized context, but not inserted into [1].

2 Estimates for the minimal eigenvalues

In this subsection we provide estimates for the minimal eigenvalues of $M_{n,p}^{{\mathbb{Q}}_{\mu}}$ and $K_{n,p}^{{\mathbb{Q}}_{\mu}}$. These estimates will be employed to obtain a lower bound for $|\lambda_{\min}(A_{n,p}^{{\mathbb{Q}}_{\mu}})|$, where $\lambda_{\min}(A_{n,p}^{{\mathbb{Q}}_{\mu}})$ is an eigenvalue of $A_{n,p}^{{\mathbb{Q}}_{\mu}}$ with minimum modulus.

We begin by generalizing [2, Eq. (51)]. We remember that the result declares how, for every $p\geq 1,n\geq 2$, and $\mathbf{x}=(x_{1},\ldots,x_{n+p-2})\in\mathbb{R}^{n+p-2},$, it holds

[TABLE]

where the constants $C_{p},\overline{C}_{p}>0$ do not depend on $n$ and $\mathbf{x}$ (see also [3, Eq. (6.3) and Theorem 9.27]. It is possible to infer that, for every $i=2,\ldots,n+p-1$, $x\in\hbox{int}(\hbox{supp}(N_{i,p}(x)))$, there hold

[TABLE]

where we use the notation ${\alpha}^{*}=\frac{{\alpha}}{\lfloor\frac{{\alpha}}{\pi}\rfloor+1}$ and

[TABLE]

Indeed, $N_{i,p}^{{\mathbb{Q}}_{n{\alpha}}}(x)$ and $N_{i,p}(x)$ are zero in the same points, and the zeros in their supports (at the boundaries) have the same order; furthermore, $N_{i,p}^{{\mathbb{Q}}_{\alpha}}(x)\rightarrow N_{i,p}(x)$ while $n\rightarrow\infty$.

As a consequence, with norms in $L_{2}([0,1])$

[TABLE]

and so, by setting $C_{p}^{{\mathbb{Q}}_{\alpha}}:=(K_{p}^{{\mathbb{Q}}_{\alpha}})^{2}C_{p}$, $\overline{C}_{p}^{{\mathbb{Q}}_{\alpha}}:=(\overline{K}_{p}^{{\mathbb{Q}}_{\alpha}})^{2}\overline{C}_{p}$, $C_{p}^{{\mathbb{Q}}_{[0,{\alpha}]}}:=(K_{p}^{{\mathbb{Q}}_{[0,{\alpha}]}})^{2}C_{p}$, $\overline{C}_{p}^{{\mathbb{Q}}_{[0,{\alpha}]}}:=(\overline{K}_{p}^{{\mathbb{Q}}_{[0,{\alpha}]}})^{2}\overline{C}_{p}$

[TABLE]

We also recall the Poincaré inequality in the one-dimensional setting:

[TABLE]

where $\frac{1}{\pi}$ is the best constant, see [4].

We can use (6)-(7) and (8) to prove the next theorem, by defining $C_{p}^{{\mathbb{Q}}_{\nu}}:=C_{p}^{{\mathbb{Q}}_{\alpha}}$ in the non-nested case, $C_{p}^{{\mathbb{Q}}_{\nu}}:=C_{p}^{{\mathbb{Q}}_{[0,{\alpha}]}}$ in the nested case.

Theorem 2.1**.**

Let $C_{p}^{{\mathbb{Q}}_{\nu}}\geq 0$ be the constant in (6) or (7), then for all $p\geq 2$ and $n\geq 2$ the following properties hold.

1. $\lambda_{\min}(M_{n,p}^{{\mathbb{Q}}_{\mu}})\geq C_{p}^{{\mathbb{Q}}_{\nu}}$,

2. $K_{n,p}^{{\mathbb{Q}}_{\mu}}\geq\frac{\pi^{2}}{n^{2}}M_{n,p}^{{\mathbb{Q}}_{\mu}}$ and $\lambda_{\min}(K_{n,p}^{{\mathbb{Q}}_{\mu}})\geq\frac{\pi^{2}C_{p}^{{\mathbb{Q}}_{\nu}}}{n^{2}}$.

Proof.

The proof is analogous to the one referring to the polynomial case, see [2, Theorem 8], by considering that $M_{n,p}^{{\mathbb{Q}}_{\mu}}$ and $K_{n,p}^{{\mathbb{Q}}_{\mu}}$ are still symmetric matrices, and having (6)-(8). ∎

Remark 2.2**.**

For every $p\geq 2,n\geq 2$ and $j=1,\ldots,n+p-2$, let $\lambda_{j}(K_{n,p}^{{\mathbb{Q}}_{\mu}})$ be the $j$-th smallest eigenvalue of $K_{n,p}^{{\mathbb{Q}}_{\mu}}$, that is, $\lambda_{1}(K_{n,p}^{{\mathbb{Q}}_{\mu}})\leq\ldots\lambda_{n+p-2}(K_{n,p}^{{\mathbb{Q}}_{\mu}})$. Then, we conjecture that for every $p\geq 2$ and for each fixed $j\geq 1$,

[TABLE]

There is a motivation related to the connection between $K_{n,p}^{{\mathbb{Q}}_{\mu}}$ and the problem [1, Eq. (1)], furthermore in the polynomial case it has been verified numerically for some values of $p$ and $j$; see [2, Remark 3] for details.

Theorem 2.3**.**

For all $p\geq 2$ and all $n\geq 2$, let $\lambda_{\min}(A_{n,p}^{{\mathbb{Q}}_{\mu}})$ be an eigenvalue of $A_{n,p}^{{\mathbb{Q}}_{\mu}}$ with minimum modulus. Then,

[TABLE]

with $C_{p}^{{\mathbb{Q}}_{\nu}}\geq 0$ being the same constant appearing in Theorem 2.1.

Proof.

Let us preliminary recall that, having $X^{*}$ the conjugate transposed of $X$, they hold:

[TABLE]

By the expression

[TABLE]

and recalling that $K_{n,p}^{{\mathbb{Q}}_{\mu}},M_{n,p}^{{\mathbb{Q}}_{\mu}}$ are symmetric, while $H_{n,p}^{{\mathbb{Q}}_{\mu}}$ is skew-symmetric, we infer that the real part of $A_{n,p}^{{\mathbb{Q}}_{\mu}}$ is given by

[TABLE]

Therefore, by the minimax principle and by Theorem 2.1 we obtain

[TABLE]

The result is obtained by considering that $|\lambda_{\min}(A_{n,p}^{{\mathbb{Q}}_{\mu}})|\geq\lambda_{\min}(\textup{Re}A_{n,p}^{{\mathbb{Q}}_{\mu}})$ because of the spectrum localization

[TABLE]

∎

The lower bound (10) remains bounded away from 0 for all $\gamma\geq 0$ and, in particular, for the interesting value $\gamma=0$.

3 Conditioning

In this subsection we provide a bound for the condition number

[TABLE]

Let us start with the Fan-Hoffman theorem ([5]).

Theorem 3.1**.**

Let $X\in\mathbb{C}^{m\times m}$ and let:

[TABLE]

be the singular values of $X$ and the eigenvalues of $\textup{Re}X$, respectively. Then

[TABLE]

Theorem 3.2**.**

For every $p\geq 2$ there exists a constant $\alpha_{p}>0$ such that

[TABLE]

Proof.

Fix $p\geq 2$ and $n\geq 2$. Being either symmetric or skew-symmetric, $K_{n,p}^{{\mathbb{Q}}_{\mu}},H_{n,p}^{{\mathbb{Q}}_{\mu}}$ and $M_{n,p}^{{\mathbb{Q}}_{\mu}}$ are normal matrices, and by applying [1, Lemma 7] we obtain for $\|A_{n,p}^{{\mathbb{Q}}_{\mu}}\|$ the following bound:

[TABLE]

On the other hand, being $s_{n+p-2}(A_{n,p}^{{\mathbb{Q}}_{\mu}})$ the minimum singular value of $A_{n,p}^{{\mathbb{Q}}_{\mu}}$, we have

[TABLE]

so

[TABLE]

that makes the theorem satisfied for $\alpha_{p}:=\frac{1}{C_{p}^{{\mathbb{Q}}_{\nu}}(\pi^{2}+\gamma)}[C_{p,\alpha}+|\beta|+\frac{\gamma(p+1)}{4}].$

∎

4 More preliminaries on spectral analysis

The following one is another important result regarding sequences of Toeplitz matrices.

Theorem 4.1**.**

Let $f\in L_{1}([-\pi,\pi])$ be a real-valued function, and let $m_{f}:=\operatorname{ess}\inf f,M_{f}:=\operatorname{ess}\sup f$, and suppose $m_{f}<M_{f}$. Then

$\lambda_{\min}(T_{m}(f))\searrow m_{f}$* and $\lambda_{\max}(T_{m}(f))\nearrow M_{f}$ as $m\rightarrow\infty$.*

Another result due to Parter [6] concerns the asymptotics of the $j$-th smallest eigenvalue $\lambda_{j}(T_{m}(f))$, for $j$ fixed and $m\rightarrow\infty$.

Theorem 4.2**.**

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be continuous and $2\pi$-periodic. Let $m_{f}:=\min_{\theta\in\mathbb{R}}f(\theta)=f(\theta_{\min})$ and let $\theta_{\min}$ be the unique point in $(-\pi,\pi]$ such that $f(\theta_{\min})=m_{f}$. Assume there exists $s\geq 1$ such that $f$ has $2s$ continuous derivatives in $(\theta_{\min}-\epsilon,\theta_{\min}+\epsilon)$ for some $\epsilon>0$ and $f^{(2s)}(\theta_{\min})>0$ is the first non-vanishing derivative of $f$ at $\theta_{\min}$. Finally, for every $m\geq 1$, let $\lambda_{1}(T_{m}(f))\leq\cdots\leq\lambda_{m}(T_{m}(f))$ be the eigenvalues of $T_{m}(f)$ arranged in non-decreasing order. Then, for each fixed $j\geq 1$,

[TABLE]

where $c_{s,j}>0$ is a constant depending only on $s$ and $j$.

Remark 4.3**.**

The constant $c_{s,j}$ is the $j$-th smallest eigenvalue of the boundary value problem

[TABLE]

see [6, p. 191]. Thus, we find that $c_{1,j}=j^{2}\pi^{2}$ for all $j\geq 1$.

It is also important to recall some properties of Toeplitz matrices having a two-level structure. In the next lemma, see e.g. [5], we collect some basic results concerning tensor-products matrices.

Lemma 4.4**.**

Suppose that $X\in\mathbb{C}^{m_{1}\times m_{1}}$ and $Y\in\mathbb{C}^{m_{2}\times m_{2}}$ are normal matrices with eigenvalues given by $\lambda_{1}(X),\ldots,\lambda_{m_{1}}(X)$ and $\lambda_{1}(Y),\ldots,\lambda_{m_{2}}(Y)$. Then

$X\otimes Y$* is normal and $(X\otimes Y)^{*}=X^{*}\otimes Y^{*}$;* 2. 2.

$\sigma(X\otimes Y)=\{\lambda_{i}(X)\lambda_{j}(Y):i=1,\ldots,m_{1},j=1,\ldots,m_{2}\}$; 3. 3.

$\operatorname{rank}(X\otimes Y)=\operatorname{rank}(X)\operatorname{rank}(Y);$** 4. 4.

$\|X\otimes Y\|=\|X\|\mbox{ }\|Y\|$* and $\|X\otimes Y\|_{1}=\|X\|_{1}\mbox{ }\|Y\|_{1}$.*

Note that statements 1 and 2 imply that if $X,Y$ are Hermitian then $X\otimes Y$ is Hermitian, and if $X,Y$ are Hermitian and positive definite then $X\otimes Y$ is Hermitian and positive definite.

Given a bivariate function $g:[-\pi,\pi]^{2}\rightarrow\mathbb{R}$ belonging to the space $L_{1}([-\pi,\pi]^{2})$, we can associate to $g$ a family of two-level Hermitian Toeplitz matrices $\{T_{m_{1},m_{2}}(g)\}$ parametrized by two indices $m_{1},m_{2}$ and defined blockwise for every $m_{1},m_{2}\geq 1$:

[TABLE]

where for every $k\in\mathbb{Z}$,

[TABLE]

and for every $k,l\in\mathbb{Z}$,

[TABLE]

is the $(k,l)$ Fourier coefficient of $g$. A Szegö-like result, analogous to [1, Theorem 2], holds for two-level sequences (see [7] and [8]).

Theorem 4.5**.**

Let $g\in L_{1}([-\pi,\pi]^{2})$ be a real-valued function, and let $m_{g}:=\operatorname{ess}\inf g,M_{g}:=\operatorname{ess}\sup g$, and suppose $m_{g}<M_{g}$. Then:

• •

$\sigma(T_{m_{1},m_{2}}(g))\subset(m_{g},M_{g}),\quad\forall m_{1},m_{2}\geq 1$;

• •

$\{T_{m_{1},m_{2}}(g)\}\sim_{\lambda}g$.

With the last result, we can relate tensor-products and Toeplitz matrices. Given two functions $f,h:[-\pi,\pi]\rightarrow\mathbb{R}$ in $L_{1}([-\pi,\pi])$, we can construct the tensor-product function, belonging to $L_{1}([-\pi,\pi]^{2})$:

[TABLE]

So, we can consider the three families of Hermitian Toeplitz matrices $\{T_{m_{1}}(f)\},\{T_{m_{2}}(h)\}$ and $\{T_{m_{1},m_{2}}(f\otimes h)\}$. By computing directly we obtain a commutative property between the operation of tensor-product and the Toeplitz operator.

Lemma 4.6**.**

Let $f,h\in L_{1}([-\pi,\pi])$ be real-valued functions. Then, for all $m_{1},m_{2}\geq 1$,

[TABLE]

5 More results on spectral distribution

The following results are a consequence of Theorem 4.1.

Lemma 5.1**.**

They hold:

$\lambda_{\min}(B_{n,p}^{{\mathbb{Q}}_{\alpha}})\searrow 0$* and $\lambda_{\max}(B_{n,p}^{{\mathbb{Q}}_{\alpha}})\rightarrow M_{f_{p}}$ as $n\rightarrow\infty$;* 2. 2.

for each fixed $j\geq 1$,

[TABLE]

where $\lambda_{1}(B_{n,p}^{{\mathbb{Q}}_{\alpha}})\leq\cdots\leq\lambda_{n+p-2}(B_{n,p}^{{\mathbb{Q}}_{\alpha}})$ are the eigenvalues of $B_{n,p}^{{\mathbb{Q}}_{\alpha}}$ in non-decreasing order; 3. 3.

$\lambda_{\min}(B_{n,p}^{{\mathbb{Q}}_{n\alpha}})\searrow 0$* and $\lambda_{\max}(B_{n,p}^{{\mathbb{Q}}_{n\alpha}})\nearrow M_{f_{p}^{{\mathbb{Q}}_{\alpha}}}$ as $n\rightarrow\infty$;* 4. 4.

$\lambda_{\min}(C_{n,p}^{{\mathbb{Q}}_{\alpha}})\rightarrow m_{h_{p}}$* and $\lambda_{\max}(C_{n,p}^{{\mathbb{Q}}_{\alpha}})\nearrow 1$ as $n\rightarrow\infty$;* 5. 5.

$\lambda_{\min}(C_{n,p}^{{\mathbb{Q}}_{n\alpha}})\searrow m_{h_{p}^{{\mathbb{Q}}_{\alpha}}}$* and $\lambda_{\max}(C_{n,p}^{{\mathbb{Q}}_{n\alpha}})\nearrow 1$ as $n\rightarrow\infty$.*

6 Two-dimensional linear elliptic differential problem

We can extend the problem [1, Eq. (1)] to the 2D domain $\Omega=(0,1)^{2}$, that is:

[TABLE]

with $f\in L^{2}((0,1)^{2}),\mbox{ }\beta=[\beta_{1}\mbox{ }\beta_{2}]^{T}\in\mathbb{R}^{2},\mbox{ }\gamma\geq 0$. We want to approximate the weak solution of (LABEL:eq:poly83) by using the Galerkin method [1, Eq. (3)]; $\mathcal{W}$, approximation space, is now chosen in a manner that takes account of the two-dimensional structure.

For the sake of simplicity, we suppose to work in the nested case; we briefly address also the non-nested case while considering, at the end, the main result of this Section, which will be proposed for both cases.

Also, for a better comparison with this study in the polynomial case, we use the inverse lexicographical ordering of [2, Section 5], rather than the standard lexicographical ordering used in [1, Section 6].

In light of this, we consider two univariate B-spline bases from [1, Section 4], one for every direction ($x$ and $y$): the basis $\{N_{i,p_{1}}^{{\mathbb{Q}}_{\alpha_{1}}}(x),i=1,\ldots,n_{1}+p_{1}\}$ over the knot sequence

[TABLE]

where

[TABLE]

and the basis $\{N_{i,p_{2}}^{{\mathbb{Q}}_{\alpha_{2}}}(y),i=1,\ldots,n_{2}+p_{2}\}$ over the knot sequence

[TABLE]

where

[TABLE]

With these, we construct our bivariate tensor-product B-spline basis $\{N_{(i,j),(p_{1},p_{2})}^{{\boldsymbol{{\mathbb{Q}}}}_{{\boldsymbol{\alpha}}}},i=1,\ldots,n_{1}+p_{1},j=1,\ldots,n_{2}+p_{2}\}$, like

[TABLE]

The space $\mathcal{W}$ in the Galerkin problem [1, Eq. (3)] is now chosen as $\mathcal{W}_{(n_{1},n_{2}),(p_{1},p_{2})}^{{\boldsymbol{{\mathbb{Q}}}}_{{\boldsymbol{\alpha}}}}$, where

[TABLE]

We order lexicographically the element of the basis (16), as follows

[TABLE]

with $i=1,\ldots,n_{1}+p_{1}-2,\mbox{ }j=1,\ldots,n_{2}+p_{2}-2$.

Once we have fixed the basis (16) and the order (17), we devise a linear system [1, Eq. (4)] from the Galerkin problem [1, Eq. (3)]. The stiffness matrix $A$ in [1, Eq. (4)] depends from $n_{1},n_{2}$ and $p_{1},p_{2}$, and can be therefore denoted by:

[TABLE]

where $a(u,v)=\int_{0}^{1}\int_{0}^{1}\nabla u\cdot\nabla v\mbox{ }dxdy+\beta\int_{0}^{1}\int_{0}^{1}\nabla u\mbox{ }v\mbox{ }dxdy+\gamma\int_{0}^{1}\int_{0}^{1}uv\mbox{ }dxdy$, see [1, Section 2.1.1]

6.1 Construction of the matrices $A_{(n_{1},n_{2}),(p_{1},p_{2})}^{{\mathbb{Q}}_{{\boldsymbol{\alpha}}}}$

Using the same integration rules of the one-dimensional case, we obtain

[TABLE]

where

[TABLE]

$M_{\bf n,p}^{{\boldsymbol{{\mathbb{Q}}}}_{{\boldsymbol{\alpha}}}}:=M_{n_{2},p_{2}}^{{\mathbb{Q}}_{\alpha_{2}}}\otimes M_{n_{1},p_{1}}^{{\mathbb{Q}}_{\alpha_{1}}}$.

A simplifying case is given by $n_{1}=n_{2}=n$ and $p_{1}=p_{2}=p$, for which, by defining $K_{\bf n,p}^{{\boldsymbol{{\mathbb{Q}}}}_{{\boldsymbol{\alpha}}}}:=\widehat{K}_{\bf n,p}^{{\boldsymbol{{\mathbb{Q}}}}_{{\boldsymbol{\alpha}}}}+\widetilde{K}_{\bf n,p}^{{\boldsymbol{{\mathbb{Q}}}}_{{\boldsymbol{\alpha}}}}$

[TABLE]

6.2 Spectral distribution

We can study now, by fixing $p_{1},p_{2}\geq 1$, the spectral distribution of the sequence of the matrices (18). We mildly assume that the ratio $\frac{n_{2}}{n_{1}}=:\nu$ is constant as $n_{1}\rightarrow\infty$; in this way, $A_{\bf n,p}^{{\boldsymbol{{\mathbb{Q}}}}_{{\boldsymbol{\alpha}}}}$ is an actual sequence of matrices, since only $n_{1}$ is a free parameter (we can set for $n_{2}$ the integer value for which the ratio $\nu$ is better approximated).

This condition could be released in favor of some lighter requests, but for the sake of simplicity (especially in notation) we choose to work in this framework, which is ultimately not particulary restrictive.

[TABLE]

For every $n_{1}\geq 3p_{1}+1$ such that $n_{2}=\nu n_{1}\geq 3p_{2}+1$, we decompose the matrices $\widehat{K}_{\bf n,p}^{{\boldsymbol{{\mathbb{Q}}}}_{{\boldsymbol{\alpha}}}}$ and $\widetilde{K}_{\bf n,p}^{{\boldsymbol{{\mathbb{Q}}}}_{{\boldsymbol{\alpha}}}}$ into

[TABLE]

where

[TABLE]

and

[TABLE]

where the matrices $B_{n_{i},p_{i}}^{{\mathbb{Q}}_{\alpha_{i}}},R_{n_{i},p_{i}}^{{\mathbb{Q}}_{\alpha_{i}}},C_{n_{i},p_{i}}^{{\mathbb{Q}}_{\alpha_{i}}},S_{n_{i},p_{i}}^{{\mathbb{Q}}_{\alpha_{i}}}$, $i=1,2$, are those of [1, Eq. (67)] and [1, Eq. (75)]. We further define:

[TABLE]

Lemmas [1, Lemma 6] and [1, Lemma 10] state that $B_{n_{i},p_{i}}^{{\mathbb{Q}}_{\alpha_{i}}}=T_{n_{i}+p_{i}-2}(f_{p_{i}}^{{\mathbb{Q}}_{\alpha_{i}/n_{i}}})$ and $C_{n_{i},p_{i}}^{{\mathbb{Q}}_{\alpha_{i}}}=T_{n_{i}+p_{i}-2}(h_{p_{i}}^{{\mathbb{Q}}_{\alpha_{i}/n_{i}}})$, for $i=1,2$, $p_{i}\geq 1$ and $n_{i}\geq 3p_{i}+1$, thanks to which by Lemma 4.6 we obtain

[TABLE]

and, by taking account of the linearity of the operator $T_{n_{2}+p_{2}-2,n_{1}+p_{1}-2}$:

[TABLE]

We can use now Theorem 4.5 to state

[TABLE]

and

[TABLE]

By Lemma 4.4 and the inequalities [1, Eq. (68)] and [1, Eq. (76)], we have

[TABLE]

and an analogous result holds for $\widetilde{R}_{\bf n,p}^{{\boldsymbol{{\mathbb{Q}}}}_{{\boldsymbol{\alpha}}}}$ , that is, as $n_{1}\rightarrow\infty$

[TABLE]

Combining the two previous results gives, again for $n_{1}\rightarrow\infty$

[TABLE]

Note that $(n_{1}+p_{1}-2)(\nu n_{1}+p_{2}-2)$ is the dimension of the matrix $A_{\bf n,p}^{{\boldsymbol{{\mathbb{Q}}}}_{{\boldsymbol{\alpha}}}}$.

The use of [1, Lemmas 6 and 8-11], along with Lemmas 4.4 and 5.1, and the fact that the matrices $K_{n_{i},p_{i}}^{{\mathbb{Q}}_{\alpha_{i}}}$, $H_{n_{i},p_{i}}^{{\mathbb{Q}}_{\alpha_{i}}}$, $M_{n_{i},p_{i}}^{{\mathbb{Q}}_{\alpha_{i}}}$, $B_{n_{i},p_{i}}^{{\mathbb{Q}}_{\alpha_{i}}}$, $C_{n_{i},p_{i}}^{{\mathbb{Q}}_{\alpha_{i}}}$, $i=1,2$, are normal, yield

[TABLE]

[1, Lemma 7] implies the existence of a constant $Q_{p_{1},p_{2}}$, independent of $n_{1}$, for which

[TABLE]

As a result, we can state a theorem which could be seen as the two-dimensional version of [1, Theorem 5].

Theorem 6.1**.**

The sequence of matrices $\{A_{\bf n,p}^{{\boldsymbol{{\mathbb{Q}}}}_{{\boldsymbol{\alpha}}}}\}$ is distributed, in the sense of the eigenvalues, like the function $g_{p_{1},p_{2}}:[-\pi,\pi]^{2}\rightarrow\mathbb{R}$

[TABLE]

Proof.

The proof is analogous to [2, Theorem 18], since matrices and functions have the same properties, and the same relations hold between them. ∎

In the non-nested case, Theorem 6.1 can be rendered as:

Theorem 6.2**.**

The sequence of matrices $\{A_{\bf n,p}^{{\boldsymbol{{\mathbb{Q}}}}_{{\boldsymbol{n}}{\boldsymbol{\alpha}}}}\}$ is distributed, in the sense of the eigenvalues, like the function $g_{p_{1},p_{2}}^{{\mathbb{Q}}_{\alpha}}:[-\pi,\pi]^{2}\rightarrow\mathbb{R}$

[TABLE]