Contraction and Optimality Properties of an Adaptive Legendre–Galerkin Method: The Multi-Dimensional Case

Abstract

We analyze the theoretical properties of an adaptive Legendre–Galerkin method in the multidimensional case. After the recent investigations for Fourier–Galerkin methods in a periodic box and for Legendre–Galerkin methods in the one dimensional setting, the present study represents a further step towards a mathematically rigorous understanding of adaptive spectral/$hp$ discretizations of elliptic boundary-value problems. The main contribution of the paper is a careful construction of a multidimensional Riesz basis in $H^1$, based on a quasi-orthonormalization procedure. This allows us to design an adaptive algorithm, to prove its convergence by a contraction argument, and to discuss its optimality properties (in the sense of non-linear approximation theory) in certain sparsity classes of Gevrey type.

Appendix

Proof

(Proof of Lemma 2 ) In the following we will make extensive use of the following property of the product of univariate Legendre polynomials (see, e.g., [1]):

$$\begin{aligned} L_m(x_i)L_n(x_i)=\sum _{r=0}^{\min (m,n)} A_{m,n}^r L_{m+n-2r} (x_i)\qquad i=1,2 \end{aligned}$$

(5.1)

with

$$\begin{aligned} A_{m,n}^r:= \frac{A_{m-r} A_r A_{n-r}}{A_{n+m-r}}\frac{2n+2m-4r+1}{2n+2m-2r+1} \end{aligned}$$

and

$$\begin{aligned} A_0:=1\ , \qquad A_m:=\frac{1\cdot 3 \cdot 5 \ldots (2m-1)}{m !}=\frac{(2m)!}{2^m (m!)^2}. \end{aligned}$$

Moreover we recall the following asymptotic estimates (see [9]):

• Case $0< r <\min (m,n)$:

  $$\begin{aligned} \frac{A_{m-r} A_r A_{n-r}}{A_{n+m-r}}\sim \frac{1}{\pi } \frac{\sqrt{n+m-r}}{\sqrt{m-r}\sqrt{n-r}\sqrt{r}}\ ; \end{aligned}$$

  (5.2)

• Case $r=0$:

  $$\begin{aligned} \frac{A_{m} A_{n}}{A_{n+m}}\sim \frac{1}{\sqrt{\pi }} \frac{\sqrt{n+m}}{\sqrt{nm}} \ ; \end{aligned}$$

  (5.3)

• Case $r=\min (m,n)$ and $m\not = n$:

  $$\begin{aligned} \frac{A_{\min (m,n)} A_{\vert m -n \vert }}{{{A_{\max (m,n)}}}}\sim \frac{1}{\sqrt{\pi }} \frac{\sqrt{\max (m,n)}}{\sqrt{\min (m,n)}\sqrt{\vert m -n \vert }}. \end{aligned}$$

  (5.4)

  When $m=n$ it is sufficient to use $A_0=1$ to get $\frac{A_{m} A_{0}}{A_{m}}=1$.

We begin from the following expression:

$$\begin{aligned} a^\eta _{mn}=\int _{\varOmega } \nu (x)\nabla \eta _m(x)\nabla \eta _n(x) dx + \int _\varOmega \sigma (x)\eta _m(x)\eta _n(x) dx=: a_{mn}^{(1)} + a^{(0)}_{mn}. \end{aligned}$$

(5.5)

We first estimate $a^{(1)}_{mn}$. Let $m=(m_1,m_2)$ and $n=(n_1,n_2)$ then using the notation $\nu :=\nu (x_1,x_2)$ and the relation $\eta ^\prime _k(x_i)=-\sqrt{k-1/2} L_{k-1}(x_i)$, $i=1,2$, we have

$$\begin{aligned} a^{(1)}_{mn}&= \int _\varOmega \nu \eta ^\prime _{m_1}(x_1)\eta ^\prime _{n_1}(x_1)\eta _{m_2}(x_2) \eta _{m_2}(x_2) dx_1 dx_2 \nonumber \\&+ \int _\varOmega \nu \eta _{m_1}(x_1)\eta _{n_1}(x_1)\eta ^\prime _{m_2}(x_2)\eta ^\prime _{m_2}(x_2) dx_1 dx_2\nonumber \\ \!&= \!B^1_{m,n}\int _{\varOmega }\! \nu L_{m_1-1}(x_1)L_{n_1-1}(x_1)[L_{m_2-2}(x_2)\!-\!L_{m_2}(x_2)][L_{n_2-2}(x_2)\!-\!L_{n_2}(x_2)]dx_1dx_2\nonumber \\&+~B^2_{m,n}\int _{\varOmega } \nu [L_{m_1-2}(x_1)-L_{m_1}(x_1)][L_{n_1-2}(x_1)\nonumber \\&-L_{n_1}(x_2)] L_{m_2-1}(x_2)L_{n_2-1}(x_2)dx_1dx_2\nonumber \\&=: J_1 + J_2 \end{aligned}$$

(5.6)

where we set

$$\begin{aligned} B^1_{m,n}:=\frac{\sqrt{m_1-1/2}\sqrt{n_1-1/2}}{\sqrt{4m_2-2} \sqrt{4n_2-2}}\qquad B^2_{m,n}:= \frac{\sqrt{m_2-1/2}\sqrt{n_2-1/2}}{\sqrt{4m_1-2}\sqrt{4n_1-2}}. \end{aligned}$$

Let us focus on the first term $J_1$. Straightforward calculations yield

$$\begin{aligned} J_1&= B^1_{m,n}\Big \{ \int _{\varOmega } \nu L_{m_1-1}(x_1)L_{n_1-1}(x_1) L_{m_2-2}(x_2)L_{n_2-2}(x_2) dx_1dx_2\nonumber \\&- \int _{\varOmega } \nu L_{m_1-1}(x_1)L_{n_1-1}(x_1) L_{m_2}(x_2)L_{n_2-2}(x_2) dx_1dx_2\nonumber \\&- \int _{\varOmega } \nu L_{m_1-1}(x_1)L_{n_1-1}(x_1) L_{m_2-2}(x_2)L_{n_2}(x_2) dx_1dx_2\nonumber \\&+ \int _{\varOmega } \nu L_{m_1-1}(x_1)L_{n_1-1}(x_1) L_{m_2}(x_2)L_{n_2}(x_2) dx_1dx_2\Big \}\nonumber \\&=: J_1^1+ J_1^2+ J_1^3+ J_1^4. \end{aligned}$$

For the ease of presentation we only show how to estimate $J_1^1$ as the other terms can be worked out similarly. Employing (5.1) we obtain

$$\begin{aligned} J_1^1&= B_{m,n}^1 \int _\varOmega \nu \sum _{r_1=0}^{\min (m_1-1,n_1-1)} A_{m_1-1,n_1-1}^{r_1} L_{m_1+n_1-2-2r_1}(x_1) \\&\qquad \sum _{r_2=0}^{\min (m_2-2,n_2-2)} A_{m_2-2,n_2-2}^{r_2} L_{m_2+n_2-4-2r_2}(x_2) ~dx_1x_2 \\&= B_{m,n}^1 \sum _{r_1=0}^{\min (m_1-1,n_1-1)} \sum _{r_2=0}^{\min (m_2-2,n_2-2)} A_{m_1-1,n_1-1}^{r_1} A_{m_2-2,n_2-2}^{r_2}\\&\int _\varOmega \nu L_{m_1+n_1-2-2r_1}L_{m_2+n_2-4-2r_2}dx_1x_2. \end{aligned}$$

Using the multidimensional Legendre expansion $\nu (x)=\sum _{k\in \mathcal{K}} \nu _k L_k(x)$ we obtain

$$\begin{aligned} J_1^1\!=\!4 B_{m,n}^1 \sum _{r_1=0}^{\min (m_1-1,n_1-1)} \sum _{r_2=0}^{\min (m_2-2,n_2-2)} \frac{A_{m_1-1,n_1-1}^{r_1} A_{m_2-2,n_2-2}^{r_2} \nu _{m_1+n_1-2-2r_1,m_2+n_2-4-2r_2}}{[2(m_1 +n_1-2-2r_1)+1][2(m_2+n_2-4-2r_2)+1]}. \end{aligned}$$

We now employ the asymptotic estimates (5.2)–(5.4) to bound the terms $A_{m_1-2,n_1-2}^{r_1}$ and $A_{m_2-2,n_2-2}^{r_2}$. Accordingly, we need to distinguish among several cases depending on the combination of the values assumed by $r_1$ and $r_2$. However, for the ease of reading, we only consider the case $0<r_1<\min (m_1-2,n_1-2)$ and $0<r_2<\min (m_2-2,n_2-2)$ as the other ones can be treated similarly. In this case, (5.2) yields

$$\begin{aligned} A_{m_1-2,n_1-2}^{r_1}\simeq \frac{1}{\pi } \frac{\sqrt{n_1+m_1-4-r_1}}{\sqrt{m_1-2-r_1}\sqrt{n_1-2-r_1}\sqrt{r_1}} \frac{2(n_1-2)+2(m_1-2)-4r_1+1}{2(n_1-2)+2(m_1-2)-2r_1+1}. \end{aligned}$$

(5.7)

A similar estimate holds also for $A_{m_2-2,n_2-2}^{r_2}$. Thus we have

$$\begin{aligned}&\frac{ B_{m,n}^1 A_{m_1-1,n_1-1}^{r_1} A_{m_2-2,n_2-2}^{r_2}}{[2(m_1+n_1-2-2r_1)+1][2(m_2+n_2-4-2r_2)+1]} \\&\quad \simeq \frac{\sqrt{2m_1-1}\sqrt{2n_1-1}}{\sqrt{2m_2-1}\sqrt{2n_2-1}} \frac{\sqrt{n_2+m_2-4-r_2}\sqrt{n_1+m_1-4-r_1} }{\sqrt{m_2-2-r_2}\sqrt{n_2-2-2r_2}\sqrt{r_2} \sqrt{m_2-2-r_2}\sqrt{n_2-2-2r_2}\sqrt{r_2}} \\&\qquad \times \frac{1}{(2m_2+2n_2-2r_2-7)(2m_1+2n_1-2r_1-3)}\\&\quad \simeq \frac{\sqrt{m_1n_1}}{\sqrt{m_1-1-r_1}\sqrt{n_1-1}\sqrt{r_1}} \frac{1}{\sqrt{n_1+m_1-2-r_1}} \frac{1}{\sqrt{n_2+m_2-4-r_2}} \frac{1}{\sqrt{2m_2-1}\sqrt{2n_2-1}}\\&\qquad \times \frac{1}{\sqrt{m_2-2-r_2}\sqrt{n_2-2-2r_2}\sqrt{r_2}}\\&\quad \simeq \frac{1}{\sqrt{\min (m_1+n_1-2,\vert m_1-n_1\vert )}} \frac{1}{\sqrt{n_2+m_2-4-r_2}} \frac{1}{\sqrt{2m_2-1}\sqrt{2n_2-1}} \\&\qquad \times \frac{1}{\sqrt{m_2-2-r_2}\sqrt{n_2-2-2r_2}\sqrt{r_2}}\\&\quad \lesssim 1. \end{aligned}$$

Thus we have

$$\begin{aligned} J_1^1 \lesssim \sum _{r_1=0}^{\min (m_1-1,n_1-1)} \sum _{r_2=0}^{\min (m_2-2,n_2-2)} \nu _{m_1+n_1-2-2r_1,m_2+n_2-4-2r_2}. \end{aligned}$$

(5.8)

Similar estimates can be obtained for the terms $J_1^2,\ldots ,J_1^4$ yielding

$$\begin{aligned} J_1&\lesssim \sum _{r_1=0}^{\min (m_1-1,n_1-1)} \sum _{r_2=0}^{\min (m_2-2,n_2-2)}\nu _{m_1+n_1-2-2r_1,m_2+n_2-4-2r_2}\\&+ \sum _{r_1=0}^{\min (m_1-1,n_1-1)} \sum _{r_2=0}^{\min (m_2,n_2-2)} \nu _{m_1+n_1-2-2r_1,m_2+n_2-2-2r_2}\\&+ \sum _{r_1=0}^{\min (m_1-1,n_1-1)} \sum _{r_2=0}^{\min (m_2-2,n_2)} \nu _{m_1+n_1-2-2r_1,m_2+n_2-2-2r_2}\\&+ \sum _{r_1=0}^{\min (m_1-1,n_1-1)} \sum _{r_2=0}^{\min (m_2,n_2)} \nu _{m_1+n_1-2-2r_1,m_2+n_2-2r_2}. \end{aligned}$$

Anaologously, we can prove the following estimate for the term $J_2$

$$\begin{aligned} J_2&\lesssim \sum _{r_1=0}^{\min (m_1-2,n_1-2)} \sum _{r_2=0}^{\min (m_2-1,n_2-1)} \nu _{m_1+n_1-4-2r_1,m_2+n_2-2-2r_2} \\&+ \sum _{r_1=0}^{\min (m_1,n_1-2)} \sum _{r_2=0}^{\min (m_2-1,n_2-1)} \nu _{m_1+n_1-2-2r_1,m_2+n_2-2-2r_2} \\&+ \sum _{r_1=0}^{\min (m_1-2,n_1)} \sum _{r_2=0}^{\min (m_2-1,n_2-1)} \nu _{m_1+n_1-2-2r_1,m_2+n_2-2-2r_2} \\&+ \sum _{r_1=0}^{\min (m_1,n_1)} \sum _{r_2=0}^{\min (m_2-1,n_2-1)} \nu _{m_1+n_1-2r_1,m_2+n_2-2-2r_2}. \end{aligned}$$

Assuming $\vert \nu _k \vert \le C_\eta e^{-\gamma \Vert k\Vert _{\ell ^{1}}}$ for every $k\in \mathcal{K}$ and employing the above estimates for $J_1$ and $J_2$, we obtain

$$\begin{aligned} \vert a_{mn}^{(1)}\vert&\lesssim e^{-\gamma (\vert m_1-n_1\vert + \vert m_2-n_2\vert )}\\&\times \left\{ \sum _{r_1=0}^{\min (m_1-1,n_1-1)} \sum _{r_2=0}^{\min (m_2-2,n_2-2)} e^{-2\gamma (\min (m_1-1,n_1-1)-r_1)} e^{-2\gamma (\min (m_2-2,n_2-2)-r_2)}\right. \\&\left. +\cdots + \sum _{r_1=0}^{\min (m_1,n_1)} \sum _{r_2=0}^{\min (m_2-1,n_2-1)} e^{-2\gamma (\min (m_1,n_1)-r_1)} e^{-2\gamma (\min (m_2-1,n_2-1)-r_2)}\right\} \\&\lesssim e^{-\gamma (\vert m_1-n_1\vert + \vert m_2-n_2\vert )}= C e^{-\gamma \Vert m-n\Vert _{\ell ^{1}}}. \end{aligned}$$

We now estimate $a^{(0)}_{mn}$. Let $m=(m_1,m_2)$ and $n=(n_1,n_2)$ then recalling (2.2) and using the notation $\sigma :=\sigma (x_1,x_2)$ we have

$$\begin{aligned} a^{(0)}_{mn}&= \int _\varOmega \sigma \eta _{m_1}(x_1)\eta _{n_1}(x_1)\eta _{m_2}(x_2)\eta _{m_2}(x_2)dx_1 dx_2 \\&= C_m^n \int _\varOmega \sigma [(L_{m_1-2}-L_{m_1}) (L_{n_1-2}-L_{n_1})](x_1)[(L_{m_2-2}-L_{m_2})(L_{n_2-2}-L_{n_2})](x_2) dx_1 dx_2 \\&= C_m^n \int _\varOmega \sigma [L_{m_1-2}L_{n_1-2}-L_{m_1-2}L_{n_1}-L_{m_1}L_{n_1-2}+L_{n_1}L_{m_1}](x_1)[L_{m_2-2}L_{n_2-2} + \\&\qquad \qquad -L_{m_2-2}L_{n_2}-L_{m_2}L_{n_2-2}+L_{n_2}L_{m_2}](x_2) dx_1 dx_2 \end{aligned}$$

with $C_m^n:=\frac{1}{\sqrt{(4m_1-1)(4m_2-1)(4n_1-1)(4n_2-1)}}$. Now employing (5.1) we obtain

$$\begin{aligned} a^{(0)}_{mn}&= C_m^n \int _\varOmega \sigma \left[ \sum _{r_1=0}^{\min (m_1-2,n_1-2)} A_{m_1-2,n_1-2}^{r_1} L_{m_1+n_1-4-2r_1} - \sum _{r_1=0}^{\min (m_1-2,n_1)} A_{m_1-2,n_1}^{r_1} L_{m_1+n_1-2-2r_1} \right. \\&\left. -\sum _{r_1=0}^{\min (m_1,n_1-2)} A_{m_1,n_1-2}^{r_1} L_{m_1+n_1-2-2r_1} +\sum _{r_1=0}^{\min (m_1,n_1)} A_{m_1,n_1}^{r_1} L_{m_1+n_1-2r_1} \right] (x_1) \\&\left[ \sum _{r_2=0}^{\min (m_2-2,n_2-2)} A_{m_2-2,n_2-2}^{r_2} L_{m_2+n_2-4-2r_2} - \sum _{r_2=0}^{\min (m_2-2,n_2)} A_{m_2-2,n_2}^{r_2} L_{m_2+n_2-2-2r_2} \right. \\&\left. -\sum _{r_2=0}^{\min (m_2,n_2-2)} A_{m_2,n_2-2}^{r_2} L_{m_2+n_2-2-2r_2} +\sum _{r_2=0}^{\min (m_2,n_2)} A_{m_2,n_2}^{r_2} L_{m_2+n_2-2r_2} \right] (x_2) dx_1dx_2. \\&= I_1+\cdots +I_{16}. \end{aligned}$$

We now need to estimate $I_1,\ldots ,I_{16}$. To simplify the exposition, we only show how to estimate $I_1$, as the other terms can be treated similarly.

Using the multidimensional Legendre expansion $\sigma (x)=\sum _{k\in \mathcal{K}} \sigma _k L_k(x)$ together with (2.1) we get

$$\begin{aligned} I_1\!&= \!C_m^n \sum _{r_1=0}^{\min (m_1-2,n_1-2)} \sum _{r_2=0}^{\min (m_2-2,n_2-2)} A_{m_1-2,n_1-2}^{r_1} A_{m_2-2,n_2-2}^{r_2} \int _\varOmega \sigma L_{m_1+n_1-4-2r_1}L_{m_2+n_2-4-2r_2} dx_1dx_2 \\&= C_m^n \sum _{r_1=0}^{\min (m_1-2,n_1-2)} \sum _{r_2=0}^{\min (m_2-2,n_2-2)}\frac{A_{m_1-2,n_1-2}^{r_1} A_{m_2-2,n_2-2}^{r_2} \sigma _{m_1+n_1-4-2r_1,m_2+n_2-4-2r_2} }{[2(m_1+n_1-4-2r_1)+1][2(m_2+n_2-4-2r_2)+1]}. \end{aligned}$$

We now employ the asymptotic estimates (5.2)–(5.4) to bound the terms $A_{m_1-2,n_1-2}^{r_1}$ and $A_{m_2-2,n_2-2}^{r_2}$. Accordingly, we need to distinguish among several cases depending on the combination of the values assumed by $r_1$ and $r_2$. However, for the ease of reading, we only consider the case $0<r_1<\min (m_1-2,n_1-2)$ and $0<r_2<\min (m_2-2,n_2-2)$ as the other ones can be treated similarly. In this case, (5.2) yields

$$\begin{aligned} A_{m_1-2,n_1-2}^{r_1}\simeq \frac{1}{\pi } \frac{\sqrt{n_1+m_1-4-r_1}}{\sqrt{m_1-2-r_1}\sqrt{n_1-2-r_1}\sqrt{r_1}} \frac{2(n_1-2)+2(m_1-2)-4r_1+1}{2(n_1-2)+2(m_1-2)-2r_1+1}. \end{aligned}$$

(5.9)

A similar estimate holds also for $A_{m_2-2,n_2-2}^{r_2}$. Hence, we have

$$\begin{aligned} \frac{C_m^n A_{m_1-2,n_1-2}^{r_1} A_{m_2-2,n_2-2}^{r_2}}{[2(m_1+n_1-4-2r_1)+1][2(m_2+n_2-4-2r_2)+1]} \lesssim 1. \end{aligned}$$

(5.10)

Employing (5.3) and (5.4) yields similar estimates for the cases $r_1=0,\min (m_1-2,n_1-2)$ and $r_2=0,\min (m_2-2,n_2-2)$. In conclusion, we get

$$\begin{aligned} I_1\lesssim \sum _{r_1=0}^{\min (m_1-2,n_1-2)} \sum _{r_2=0}^{\min (m_2-2,n_2-2)}\sigma _{m_1+n_1-4-2r_1,m_2+n_2-4-2r_2}. \end{aligned}$$

(5.11)

Similar estimates can be obtained for $I_2,\ldots ,I_{16}$ thus yielding

$$\begin{aligned} a_{mn}^{(0)}&\lesssim \sum _{r_1=0}^{\min (m_1-2,n_1-2)} \sum _{r_2=0}^{\min (m_2-2,n_2-2)} \sigma _{m_1+n_1-4-2r_1,m_2+n_2-4-2r_2} \\&+ \cdots + \sum _{r_1=0}^{\min (m_1,n_1)} \sum _{r_2=0}^{\min (m_2,n_2)}\sigma _{m_1+n_1-2r_1,m_2+n_2-2r_2}. \end{aligned}$$

Assuming $\vert \sigma _k \vert \le C_\eta e^{-\gamma \Vert k\Vert _{\ell ^{1}}}$ for every $k\in \mathcal{K}$ and employing the above estimate, we obtain

$$\begin{aligned} \vert a_{mn}^{(0)}\vert&\lesssim e^{-\gamma (\vert m_1-n_1\vert + \vert m_2-n_2\vert )}\\&\times \left\{ \sum _{r_1=0}^{\min (m_1-2,n_1-2)} \sum _{r_2=0}^{\min (m_2-2,n_2-2)} e^{-2\gamma (\min (m_1-2,n_1-2)-r_1)} e^{-2\gamma (\min (m_2-2,n_2-2)-r_2)}\right. \\&\left. + \cdots + \sum _{r_1=0}^{\min (m_1,n_1)} \sum _{r_2=0}^{\min (m_2,n_2)} e^{-2\gamma (\min (m_1,n_1)-r_1)} e^{-2\gamma (\min (m_2,n_2)-r_2)}\right\} \\&\lesssim e^{-\gamma (\vert m_1-n_1\vert + \vert m_2-n_2\vert )}= C e^{-\gamma \Vert m-n\Vert _{\ell ^{1}}}. \end{aligned}$$

This concludes the proof. $\square$