In this section we derive three different methods to approximate problem (1). In all three methods we exploit the special structure of the domain Ωℓ and the right-hand side F. Our goal is to reduce the original d-dimensional problem on Ωℓ to one or more (d − 1)-dimensional problems on ω. Compared to standard methods like finite elements methods or finite difference methods, which solve the equations on Ωℓ, this strategy can significantly reduce the computational cost since ℓ is considered large and the discretisation in the x1 direction can be avoided.
Method 1: A One-term Approximation Based on an Asymptotic Analysis of Problem (1)
Although the right-hand side F in (1) is independent of x1, it is easy to see that this is not the case for the solution uℓ, i.e., due to the homogeneous Dirichlet boundary conditions it is clear that uℓ depends on x1. However, if ℓ is large one can expect that uℓ is approximately constant with respect to x1 in a subdomain \({{\varOmega }}_{\ell _{0}}\), where 0 < ℓ0 ≪ ℓ and thus converges locally to a function independent of x1 for \(\ell \rightarrow \infty \). The asymptotic behaviour of the solution uℓ when \(\ell \rightarrow \infty \) has been investigated in [8]. It can be shown that
$$ u_{\ell}\longrightarrow1\otimes u_{\infty}\qquad\text{in }{{\varOmega}}_{\ell_{0}}, $$
where \(u_{\infty }\) is the solution of (3), with an exponential rate of convergence. More precisely, the following theorem holds:
Theorem 1
There exist constants c,α > 0 independent of ℓ s.t.
$$ {\int}_{{{\varOmega}}_{\ell/2}}|\nabla(u_{\ell}-1\otimes u_{\infty})|^{2}dx \leq c\mathrm{e}^{-\alpha\ell}\|f\|_{2,\omega}^{2}, $$
where ∥⋅∥2,ω refers to the L2(ω)-norm.
For a proof we refer to [8, Theorem 6.6].
Theorem 1 shows that \(1\otimes u_{\infty }\) is a good approximation of uℓ in Ωℓ/2 when ℓ is large. This motivates to seek approximations of uℓ in Ωℓ which are of the form
$$ u_{\ell}\approx u_{\ell}^{M_{1}}:=\psi_{\ell}\otimes u_{\infty}, $$
where \(\psi _{\ell }\in {H_{0}^{1}}(-\ell ,\ell )\). Here, we choose ψℓ to be the solution of the following best approximation problem: Given \(u_{\ell }\in {H_{0}^{1}}({{\varOmega }}_{\ell })\) and \(u_{\infty }\in {H_{0}^{1}}(\omega )\), find \(\psi \in {H_{0}^{1}}(-\ell ,\ell )\) s.t.
$$ \|\nabla(u_{\ell}-\psi\otimes u_{\infty})\|_{2} = \underset{\theta\in {H_{0}^{1}}(-\ell,\ell)}{\inf}\|\nabla(u_{\ell}-\theta\otimes u_{\infty})\|_{2}. $$
(4)
In order to solve problem (4) we define the functional
$$ J(u_{\ell},u_{\infty})(\theta):=\|\nabla(u_{\ell}-\theta\otimes u_{\infty})\|_{2}^{2} $$
and consider the variational problem of minimizing it with respect to \(\theta \in {H_{0}^{1}}(-\ell ,\ell )\).
A simple computation shows that this is equivalent to finding \(\tilde {\theta }\in {H_{0}^{1}}(I_{\ell })\) such that
$$ \left. \begin{array}{ll} &\left( \nabla\left( \theta\otimes u_{\infty}\right),\nabla\left( \tilde{\theta}\otimes u_{\infty}\right)\right)_{2}=\left( \nabla u_{\ell},\nabla\left( \tilde{\theta}\otimes u_{\infty}\right)\right)_{2}\\ \Longleftrightarrow \quad& \left( \left( \begin{array}{c} \theta^{\prime}\otimes u_{\infty}\\ \theta\otimes\nabla^{\prime}u_{\infty} \end{array}\right), \left( \begin{array}{c} \tilde{\theta}^{\prime}\otimes u_{\infty}\\ \tilde{\theta}\otimes\nabla^{\prime}u_{\infty} \end{array}\right)\right)_{2}=\left( -{{\varDelta}} u_{\ell},\tilde{\theta}\otimes u_{\infty}\right)_{2}\\ \Longleftrightarrow \quad& {\alpha}_{\infty}^{2}\left( \theta^{\prime},\tilde{\theta}^{\prime}\right)_{2,I_{\ell}}+\beta_{\infty}^{2}\left( \theta,\tilde{\theta}\right)_{2,I_{\ell}}=\left( 1\otimes f,\tilde{\theta}\otimes u_{\infty}\right)_{2}\\ \Longleftrightarrow\quad & {\alpha}_{\infty}^{2}\left( \theta^{\prime},\tilde{\theta}^{\prime}\right)_{2,I_{\ell}}+\beta_{\infty}^{2}\left( \theta,\tilde{\theta}\right)_{2,I_{\ell}}=\left( 1\otimes\left( -{{\varDelta}}^{\prime}u_{\infty}\right),\tilde{\theta}\otimes u_{\infty}\right)_{2}\\ \Longleftrightarrow\quad& {\alpha}_{\infty}^{2}\left( \theta^{\prime},\tilde{\theta}^{\prime}\right)_{2,I_{\ell}}+\beta_{\infty}^{2}\left( \theta,\tilde{\theta}\right)_{2,I_{\ell}}={\beta}_{\infty}^{2}{\int}_{I_{\ell}}\tilde{\theta}. \end{array} \right\} \quad\forall\tilde{\theta}\in {H_{0}^{1}}\left( I_{\ell}\right) $$
with
$$ \alpha_{\infty}:=\|u_{\infty}\|_{2,\omega},\quad \beta_{\infty}:=\|\nabla^{\prime}u_{\infty}\|_{2,\omega}. $$
The strong form of the resulting equation is
$$ \begin{array}{@{}rcl@{}} -\alpha_{\infty}^{2}\theta^{\prime\prime}+\beta_{\infty}^{2}\theta &=&\beta_{\infty}^{2}\quad\text{ on }(-\ell,\ell),\\ \theta(-\ell)=\theta(\ell) & =& 0. \end{array} $$
The solution of this one-dimensional boundary value problem is given by
$$ \theta(x_{1}) := 1-\frac{\cosh\left( \frac{\beta_{\infty}}{\alpha_{\infty}}x_{1}\right)}{\cosh\left( \frac{\beta_{\infty}}{\alpha_{\infty}}\ell\right)}. $$
This shows that an approximation of our original problem (1) is given by
$$ u_{\ell}^{M_{1}}:=\psi_{\ell}(\lambda_{\infty},\cdot) \otimes u_{\infty},\quad\text{ with }\psi_{\ell}(a,x_{1}):=1-\frac{\cosh(ax_{1})}{\cosh(a\ell)} $$
(5)
and
$$ \lambda_{\infty}=\frac{\beta_{\infty}}{\alpha_{\infty}}=\frac{\sqrt{(f,u_{\infty})_{2,\omega}}}{\alpha_{\infty}}. $$
Note that ψℓ(a,⋅) satisfies
$$ -\psi_{\ell}^{\prime\prime}(a,\cdot)+a^{2}\psi_{\ell}(a,\cdot) = a^{2}\quad\text{ and }\quad\psi_{\ell}(\pm\ell) = 0. $$
(6)
In Section 4 we report on various numerical experiments that show the approximation properties of this rather simple one-term approximation.
Figure 1 shows a plot of \(\psi _{\ell }(\lambda _{\infty },\cdot )\) for ℓ = 20 and \(\lambda _{\infty }=2\). Since ψℓ approaches 1 with an exponential rate as x1 moves away from ± ℓ towards the origin, an analogous result to Theorem 1 can be shown for \({u}_{\ell }^{M_{1}}\).
Lemma 2
There exist constants \(c,\tilde {c}>0\) independent of ℓ such that, for δℓ < ℓ,
$$ \|\nabla(u_{\ell}-\psi_{\ell}(\lambda_{\infty},\cdot) \otimes u_{\infty})\|_{2,{{\varOmega}}_{\ell-\delta_{\ell}}}^{2} \leq C_{\omega,\delta_{\ell}}^{(1)}\|u_{\infty}\|_{2,\omega}^{2}+C_{\omega,\delta_{\ell}}^{(2)}\|\nabla^{\prime}u_{\infty}\|_{2,\omega}^{2}, $$
(7)
with
$$ C_{\omega,\delta_{\ell}}^{(1)}:=4\mathrm{e}^{-2\lambda_{1}\delta_{\ell}},\qquad C_{\omega,\delta_{\ell}}^{(2)}:=4\left( \frac{1}{\lambda_{1}}\mathrm{e}^{-2\lambda_{1}\delta_{\ell}}+6(\ell-\delta_{\ell}) \mathrm{e}^{-2\lambda_{1}\ell}\right) $$
and
$$ \lambda_{1}:=\underset{v\in {H_{0}^{1}}(\omega)\backslash\{0\}}{\inf}\frac{\|\nabla^{\prime}v\|_{2,\omega}}{\|v\|_{2,\omega}}. $$
(8)
The right-hand side in (7) goes to 0 with an exponential rate of convergence if δℓ is bounded from below when \(\ell \rightarrow \infty \).
Proof
For i = 1,2,…, let wi be the i-th eigenfunction of \(-{{\varDelta }}^{\prime }\), i.e., \(w_{i}\in {H_{0}^{1}}(\omega )\) is a solution of
$$ (\nabla^{\prime}w_{i},\nabla^{\prime}v)_{2,\omega}={{\lambda}_{i}^{2}}(w_{i},v)_{2,\omega}\qquad\forall v\in {H_{0}^{1}}(\omega) $$
(9)
and we normalize the eigenfunctions such that (wi,wj)2,ω = δi,j and order them such that (λi)i is increasing monotonously. Furthermore let \(u_{\ell ,i}\in {{H}_{0}^{1}}({{\varOmega }}_{\ell })\) be the solution of
$$ (\nabla u_{\ell,i},\nabla v)_{2}={{\lambda}_{i}^{2}}(1\otimes w_{i},v)_{2}\qquad\forall v\in {{H}_{0}^{1}}({{\varOmega}}_{\ell}). $$
Then one concludes from (6) and (9) that
$$ u_{\ell,i}=\psi_{\ell}(\lambda_{i},\cdot) \otimes w_{i}. $$
If f ∈ L2(ω) it holds
$$ f=\sum\limits_{i=1}^{\infty}(f,w_{i})_{2,\omega}w_{i}. $$
This shows that the solutions of (3) and (1) can be expressed as
$$ \begin{array}{@{}rcl@{}} u_{\infty} & =& \sum\limits_{i=1}^{\infty}\frac{(f,w_{i})_{2,\omega}}{{\lambda_{i}^{2}}}w_{i},\\ u_{\ell} & =& \sum\limits_{i=1}^{\infty}\frac{(f,w_{i})_{2,\omega}}{{{\lambda}_{i}^{2}}}u_{\ell,i}=\sum\limits_{i=1}^{\infty}\frac{(f,w_{i})_{2,\omega}}{\lambda_{i}}\psi_{\ell}(\lambda_{i},\cdot) \otimes w_{i}. \end{array} $$
With ψℓ as in (5) we get
$$ u_{\ell}-\psi_{\ell}(\lambda_{\infty},\cdot)\otimes u_{\infty} = \sum\limits_{i=1}^{\infty}\frac{(f,w_{i})_{2,\omega}}{{\lambda_{i}^{2}}}\phi_{\ell,i}\otimes w_{i},\qquad \phi_{\ell,i} (x_{1}):=\frac{\cosh(\lambda_{\infty}x_{1})}{\cosh(\lambda_{\infty}\ell)}-\frac{\cosh(\lambda_{i}x_{1})}{\cosh(\lambda_{i}\ell)}. $$
Let δℓ < ℓ. Then, since \({\int \limits }_{\omega }w_{i}w_{j}dx^{\prime }=\delta _{i,j}\), we get
$$ \begin{array}{@{}rcl@{}} |\nabla(u_{\ell}-\psi_{\ell}(\lambda_{\infty},\cdot)\otimes u_{\infty}) |_{2,{{\varOmega}}_{\ell-\delta_{\ell}}}^{2} &=& {\int}_{-\ell+\delta_{\ell}}^{\ell-\delta_{\ell}}{\int}_{\omega}|\nabla(u_{\ell}-\psi_{\ell}(\lambda_{\infty},\cdot) \otimes u_{\infty})|^{2}dx\\ &=&\sum\limits_{i=1}^{\infty}\frac{(f,w_{i})^{2}}{{{\lambda}_{i}^{4}}}{\int}_{-\ell + \delta_{\ell}}^{\ell-\delta_{\ell}}\left( (\phi_{\ell,i}^{\prime})^{2}+{\lambda_{i}^{2}}\phi_{\ell,i}^{2}\right). \end{array} $$
(10)
One has for any α > 0
$$ \begin{array}{@{}rcl@{}} {\int}_{-\ell+\delta_{\ell}}^{\ell-\delta_{\ell}}\left( \frac{\cosh(\alpha x_{1})}{\cosh(\alpha\ell)}\right)^{2}dx_{1} & =& 2{\int}_{0}^{\ell-\delta_{\ell}}\left( \frac{\cosh(\alpha x_{1})}{\cosh(\alpha\ell)}\right)^{2}dx_{1}\\ & =& \frac{1}{2}{\int}_{0}^{\ell-\delta_{\ell}}\frac{\mathrm{e}^{2\alpha x_{1}}+2+\mathrm{e}^{-2\alpha x_{1}}}{\cosh(\alpha\ell)^{2}}dx_{1}\\ &\leq& 2{\int}_{0}^{\ell-\delta_{\ell}}\frac{\operatorname{e}^{2\alpha x_{1}}+3}{\mathrm{e}^{2\alpha\ell}}dx_{1}\\ &\leq& \frac{1}{\alpha}\mathrm{e}^{-2\alpha\delta_{\ell}}+6(\ell -\delta_{\ell})\mathrm{e}^{-2\alpha\ell} \end{array} $$
and similarly
$$ {\int}_{-\ell+\delta_{\ell}}^{\ell-\delta_{\ell}}\left( \frac{\sinh(\alpha x_{1})}{\cosh(\alpha\ell)}\right)^{2}dx_{1} \leq \frac{1}{\alpha}\mathrm{e}^{-2\alpha\delta_{\ell}}. $$
Since λ1 ≤ λi for all \(i\in \mathbb {N}\) and
$$ \lambda_{1}=\underset{v\in {H_{0}^{1}}(\omega)\backslash\{0\}}{\inf}\frac{\|\nabla^{\prime}v\|_{2,\omega}}{\|v\|_{2,\omega}} \leq \frac{\|\nabla^{\prime}u_{\infty}\|_{2,\omega}}{\|u_{\infty}\|_{2,\omega}}=\lambda_{\infty}, $$
we get
$$ {\int}_{-\ell+\delta_{\ell}}^{\ell-\delta_{\ell}}\left( {\phi}_{\ell,i}^{\prime}\right)^{2} \leq 2\mathrm{e}^{-2\lambda_{\infty}\delta_{\ell}}+ 2\mathrm{e}^{-2\lambda_{1}\delta_{\ell}} \leq 4\mathrm{e}^{-2\lambda_{1}\delta_{\ell}}={C}_{\omega,\delta_{\ell}}^{(1)} $$
(11)
and
$$ \begin{array}{@{}rcl@{}} {\int}_{-\ell+\delta_{\ell}}^{\ell-\delta_{\ell}}\phi_{\ell,i}^{2} & \leq & 2{\int}_{-\ell+\delta_{\ell}}^{\ell-\delta_{\ell}}\left( \left\vert \frac{\cosh(\lambda_{\infty}x_{1})}{\cosh(\lambda_{\infty}\ell)}\right\vert^{2} + \left\vert \frac{\cosh(\lambda_{i}x_{1})}{\cosh(\lambda_{i}\ell)}\right\vert^{2}\right) \\ &\leq& 2\left( \frac{1}{\lambda_{\infty}}\mathrm{e}^{-2\lambda_{\infty}\delta_{\ell}}+\frac{1}{\lambda_{i}}\mathrm{e}^{-2\lambda_{i}\delta_{\ell}} + 6(\ell-\delta_{\ell}) \mathrm{e}^{-2\lambda_{\infty}\ell} + 6(\ell-\delta_{\ell})\mathrm{e}^{-2\lambda_{i}\ell}\right) \\ &\leq& 4\left( \frac{1}{\lambda_{1}}\mathrm{e}^{-2\lambda_{1}\delta_{\ell}}+6(\ell-\delta_{\ell}) \mathrm{e}^{-2\lambda_{1}\ell}\right) = C_{\omega,\delta_{\ell}}^{(2)}. \end{array} $$
(12)
We employ the estimates (11) and (12) in (10) and obtain
$$ \begin{array}{@{}rcl@{}} \|\nabla(u_{\ell}-\psi_{\ell}(\lambda_{\infty},\cdot) \otimes u_{\infty})\|_{2,{{\varOmega}}_{\ell-\delta_{\ell}}}^{2} & \leq& {C}_{\omega,\delta_{\ell}}^{(1)}\sum\limits_{i=1}^{\infty}\frac{(f,w_{i})_{2,\omega}^{2}}{{\lambda_{i}^{4}}}+{C}_{\omega,\delta_{\ell}}^{(2)}\sum\limits_{i=1}^{\infty}\frac{(f,w_{i})_{2,\omega}^{2}}{{\lambda_{i}^{2}}}\\ &=& C_{\omega,\delta_{\ell}}^{(1)}\|u_{\infty}\|_{2,\omega}^{2}+C_{\omega,\delta_{\ell}}^{(2)}\|\nabla^{\prime}u_{\infty}\|_{2,\omega}^{2}, \end{array} $$
which shows the assertion. □
Lemma 2 suggests that one cannot expect convergence of the approximation \(\psi _{\ell }(\lambda _{\infty },\cdot ) \otimes u_{\infty }\) on the whole domain Ωℓ. Indeed it can be shown that, in general, \(\|\nabla (u_{\ell }-\psi _{\ell }(\lambda _{\infty },\cdot )\otimes u_{\infty }) \|_{2,{{\varOmega }}_{\ell }}\nrightarrow 0\) as \(\ell \rightarrow \infty \). Setting δℓ = 0 in Lemma 2 shows that the error on Ωℓ can be estimated as follows:
Corollary 3
It holds
$$ \|\nabla(u_{\ell}-\psi_{\ell}(\lambda_{\infty},\cdot)\otimes u_{\infty})\|_{2,{{\varOmega}}_{\ell}}^{2} \leq 4\left( \|u_{\infty}\|_{2,\omega}^{2} + 6\ell\mathrm{e}^{-2\lambda_{1}\ell}\|\nabla^{\prime}u_{\infty}\|_{2,\omega}^{2}\right), $$
where λ1 is as in (8).
Method 2: An Alternating Least Squares Type Iteration
Method 1 can be interpreted as a 2-step algorithm to obtain an approximation \(u^{M_{1}}_{\ell }\) of uℓ.
-
Step 1: Solve (3) in order to obtain an approximation of the form \(1\otimes u_{\infty }\) which is non-conforming, i.e., does not belong to \({{H}_{0}^{1}}({{\varOmega }}_{\ell })\).
-
Step 2: Using \(u_{\infty }\), find a function ψℓ that satisfies (4) in order to obtain the conforming approximation \(u_{\ell }^{M_{1}}:=\psi _{\ell }(\lambda _{\infty },\cdot )\otimes u_{\infty }\in {{H}_{0}^{1}}({{\varOmega }}_{\ell })\).
In this section we extend this idea and seek approximations of the form
$$ {u}_{\ell,m}^{M_{2}}=\sum\limits_{j=0}^{m}p^{(j)}\otimes q^{(j)} $$
(13)
by iteratively solving least squares problems similar to (4). We denote by
$$ \text{ Res}_{m}=u_{\ell}-u_{\ell,m}^{M_{2}}=u_{\ell}-\sum\limits_{j=0}^{m}p^{(j)}\otimes q^{(j)} $$
the residual of the approximation and suggest the following iteration to obtain \(u_{\ell ,m}^{M_{2}}\):
-
m = 0: Set \(q^{(0)}=u_{\infty }\) and \(p^{(0)}=\psi _{\ell }(\lambda _{\infty },\cdot )\).
-
m > 0: Find \(q^{(m)}\in {{H}_{0}^{1}}(\omega )\) s.t.
$$ q^{(m)}=\underset{q\in {{H}_{0}^{1}}(\omega)}{{\arg\min}}\|\nabla\left( \text{Res}_{m-1}-p^{(m-1)}\otimes q\right)\|_{2}. $$
(14)
Then, given q(m), find \(p^{(m)}\in {{H}_{0}^{1}}(I_{\ell })\) s.t.
$$ p^{(m)}=\underset{p\in {{H}_{0}^{1}}(I_{\ell})}{{\arg\min}}\|\nabla\left( \text{Res}_{m-1}-p\otimes q^{(m)}\right)\|_{2}. $$
(15)
Iterate (14) and (15) until a stopping criterion is reached (inner iteration). Then set Resm = Resm− 1 − p(m) ⊗ q(m).
The algorithm exhibits properties of a greedy algorithm. It is easy to see that in each step of the (outer) iteration the error decreases or stays constant. We focus here on its accuracy in comparison with the two other methods via numerical experiments. We emphasize that for tensors of order at least 3, convergence can be shown for the (inner) iteration (see [11, 25, 29, 30]). This limit, however, is not a global minimum in general. The outer iteration can be shown to converge as well against the true solution uℓ under the condition that we find the best rank-1 approximation in the inner iteration (see [12]).
The idea of computing approximations in the separated form (13) by iteratively enriching the current solution with rank-1 terms is known in the literature as Proper Generalized Decomposition (PGD). The PGD has been applied to various problems in computational mechanics (e.g. [1, 4, 5, 21]), computational rheology [6], quantum chemistry (e.g. [2, 3]) and others.
An extensive review of the method can be found in [7]. For an error and convergence analysis of the (outer) iteration in the case of the Poisson equation we also refer to [20], where a similar (but not identical) approach as ours is considered.
In each step of the (outer) iteration above we need to solve at least two minimization problems (14) and (15). In the following we derive the strong formulations of these problems.
Resolution of (14)
As before an investigation of the functional
$$ J(q^{(m)}):=\|\nabla(\text{Res}_{m-1}-p^{(m-1)}\otimes q^{(m)})\|_{2}^{2} $$
shows that q(m) needs to satisfy
$$ \begin{array}{@{}rcl@{}} \left( \nabla\left( p^{(m-1)}\otimes q^{(m)}\right),\nabla\left( p^{(m-1)}\otimes q\right)\right)_{2}\!& = &\! \left( \nabla\text{Res}_{m-1},\nabla\left( p^{(m-1)}\otimes q\right)\right)_{2}\\ \Longleftrightarrow\quad p_{0,m-1}\left( -{{\varDelta}}^{\prime}q^{(m)},q\right)_{2,\omega}+ p_{1,m-1}\left( q^{(m)},q\right)_{2,\omega} &=& \left( -{{\varDelta}}\text{Res}_{m-1},p^{(m-1)}\otimes q\right)_{2} \end{array} $$
for all \(q\in {H_{0}^{1}}(\omega )\), where
$$ p_{0,m-1}:=\|p^{(m-1)}\|_{2,I_{\ell}}^{2},\quad p_{1,m-1}:=\|\left( p^{(m-1)}\right)^{\prime}\|_{2,I_{\ell}}^{2}. $$
For the right-hand side we obtain
$$ \begin{array}{@{}rcl@{}} &&\left( -{{\varDelta}}\text{Res}_{m-1},p^{(m-1)}\otimes q\right)_{2}= \left( -{{\varDelta}}\left( u_{\ell}-\sum\limits_{j=0}^{m-1}p^{(j)}\otimes q^{(j)}\right),p^{(m-1)}\otimes q\right)_{2}\\ &&\qquad=\left( 1\otimes f,p^{(m-1)}\otimes q\right)_{2} + \left( \sum\limits_{j=0}^{m-1}\left( p^{(j)}\right)^{\prime\prime}\otimes q^{(j)} + p^{(j)}\otimes{{\varDelta}}^{\prime}q^{(j)},p^{(m-1)}\otimes q\right)_{2}\\ &&\qquad=\tilde{p}_{m-1}(f,q)_{2,\omega}+\sum\limits_{j=0}^{m-1}\left( \tilde{p}_{2,j,m-1}\left( q^{(j)},q\right)_{2,\omega}+\tilde{p}_{0,j,m-1}\left( {{\varDelta}}^{\prime}q^{(j)},q\right)_{2,\omega}\right), \end{array} $$
where
$$ \tilde{p}_{m-1}:={\int}_{-\ell}^{\ell}p^{(m-1)},\quad \tilde{p}_{2,j,m-1}:=\left( \left( p^{(j)}\right)^{\prime\prime},p^{(m-1)}\right)_{2,I_{\ell}},\quad\tilde{p}_{0,j,m-1}:=\left( p^{(j)},p^{(m-1)}\right)_{2,I_{\ell}}. $$
In order to compute (14) we therefore have to solve in ω
$$ -p_{0,m-1}{{\varDelta}}^{\prime}q^{(m)}+p_{1,m-1}q^{(m)}=\tilde{p}_{m-1}f+\sum\limits_{j=0}^{m-1}\left( \tilde{p}_{2,j,m-1}q^{(j)}+\tilde{p}_{0,j,m-1}{{\varDelta}}^{\prime}q^{(j)}\right). $$
(16)
Resolution of (15)
Setting the derivative of the functional
$$ J(p^{(m)}):=\|\nabla(\text{Res}_{m-1}-p^{(m)}\otimes q^{(m)})\|_{2}^{2} $$
to zero, shows that p(m) needs to satisfy
$$ \begin{array}{@{}rcl@{}} \left( \nabla\left( p^{(m)}\otimes q^{(m)}\right),\nabla\left( p\otimes q^{(m)}\right)\right)_{2} & = & \left( \nabla\text{Res}_{m},\nabla\left( p\otimes q^{(m)}\right)\right)_{2}\\ \Longleftrightarrow\quad-q_{0,m}\left( \left( p^{(m)}\right)^{\prime\prime},p\right)_{2,I_{\ell}} + q_{1,m}\left( p^{(m)},p\right)_{2,I_{\ell}} & = & \left( -{{\varDelta}}\text{Res}_{m},p\otimes q^{(m)}\right)_{2} \end{array} $$
for all \(p\in {{H}_{0}^{1}}(-\ell ,\ell )\), where
$$ q_{0,m}=\|q^{(m)}\|_{2,\omega}^{2},\quad q_{1,m}=\|\nabla^{\prime}q^{(m)}\|_{2,\omega}^{2}. $$
For the right-hand side we obtain
$$ \begin{array}{@{}rcl@{}} &&\left( -{{\varDelta}}\text{Res}_{m-1},p\otimes q^{(m)}\right)_{2}= \left( -{{\varDelta}}\left( u_{\ell}-\sum\limits_{j=0}^{m-1}p^{(j)}\otimes q^{(j)}\right),p\otimes q^{(m)}\right)_{2}\\ &&\qquad=\left( 1\otimes f,p\otimes q^{(m)}\right)_{2} +\left( \sum\limits_{j=0}^{m-1}\left( \left( p^{(j)}\right)^{\prime\prime}\otimes q^{(j)} + p^{(j)}\otimes{{\varDelta}}^{\prime}q^{(j)}\right),p\otimes q^{(m)}\right)_{2}\\ &&\qquad=\tilde{q}_{m}{\int}_{-\ell}^{\ell}p+\sum\limits_{j=0}^{m-1}\left( \tilde{q}_{0,j,m}\left( \left( p^{(j)}\right)^{\prime\prime},p\right)_{2,I_{\ell}}+\tilde{q}_{2,j,m}\left( p^{(j)},p\right)_{2,I_{\ell}}\right), \end{array} $$
where
$$ \tilde{q}_{m}:=\left( f,q^{(m)}\right)_{2,\omega},\quad\tilde{q}_{2,j,m}:=\left( {{\varDelta}}^{\prime}q^{(j)},q^{(m)}\right)_{2,\omega},\quad\tilde{q}_{0,j,m}:=\left( q^{(j)},q^{(m)}\right)_{2,\omega}. $$
In order to obtain the solution of (15) we therefore have to solve in Iℓ
$$ -q_{0,m}\left( p^{(m)}\right)^{\prime\prime}+q_{1,m}p^{(m)}=\tilde{q}_{m}+\sum\limits_{j=0}^{m-1}\left( \tilde{q}_{2,j,m}p^{(j)}+\tilde{q}_{0,j,m}\left( p^{(j)}\right)^{\prime\prime}\right). $$
(17)
Remark 4
The constants p1,m− 1, \(\tilde {p}_{2,j,m-1}\), q1,m and \(\tilde {q}_{2,j,m}\) involve derivatives and Laplace-operators. Note that after solving (16) and (17) for q(m) and p(m), discrete versions of \({{\varDelta }}^{\prime }q^{(m)}\) and \((p^{(m)})^{\prime \prime }\) can be easily obtained via the same equations. Furthermore, since
$$ q_{1,m}=\|\nabla^{\prime}q^{(m)}\|_{2,\omega}^{2}=\left( -{{\varDelta}}^{\prime}q^{(m)},q^{(m)}\right)_{2,\omega}=-\tilde{q}_{2,m,m} $$
and
$$ p_{1,m-1}=\|\left( p^{(m-1)}\right)^{\prime}\|_{2,I_{\ell}}^{2} = \left( -\left( p^{(m-1)}\right)^{\prime\prime},p^{(m-1)}\right)_{2,I_{\ell}}=-\tilde{p}_{2,m-1,m-1} $$
a numerical computation of the gradients can be avoided.
Method 3: Exploiting the Tensor Product Structure of the Operator
In this section we exploit the tensor product structure of the Laplace operator and the domain Ωℓ. Recall that
$$ {{\varOmega}}_{\ell}=I_{\ell}\times\omega. $$
Note that we do not assume that ω has a tensor product structure. Furthermore the Laplace operator in our original problem (1) can be written as
$$ -{{\varDelta}}=-{\partial_{1}^{2}}-{{\varDelta}}^{\prime}. $$
(18)
We discretize (1) with F as in (2) on a mesh \(\mathcal {G}\), e.g., by finite elements or finite differences on a tensor mesh, i.e., each mesh cell has the form (xi− 1,xi) × τj, where τj is an element of the mesh for ω. The essential assumption is that the system matrix for the discrete version of −Δ in (18) is of the tensor form
$$ A=A_{1}\otimes M^{\prime}+M_{1}\otimes A^{\prime}. $$
(19)
If we discretize with a finite difference scheme on an equidistant grid for Iℓ with step size h, then A1 is the tridiagonal matrix h− 2tridiag[− 1,2,− 1] and M1 is the identity matrix. A finite element discretisation with piecewise linear elements leads as well to A1 = h− 2tridiag[− 1,2,− 1], while \(M_{x_{1}}=\text {tridiag}[1/6,2/3,1/6]\). It can be shown that the inverse of the matrix A can be efficiently approximated with a sum of matrix exponentials. More precisely the following Theorem holds which is proved in [14, Proposition 9.34].
Theorem 5
Let M(j), A(j) be positive definite matrices with \(\lambda _{\min \limits }^{(j)}\) and \(\lambda _{\max \limits }^{(j)}\) being the extreme eigenvalues of the generalized eigenvalue problem A(j)x = λM(j)x and set
$$ \begin{array}{@{}rcl@{}} A &= & A^{(1)}\otimes M^{(2)}\otimes\cdots\otimes M^{(n)}+M^{(1)}\otimes A^{(2)}\otimes\cdots\otimes M^{(n)}+\cdots\\ &&+M^{(1)}\otimes\cdots\otimes M^{(n-1)}\otimes A^{(n)}. \end{array} $$
Then A− 1 can be approximated by
$$ B:=\left( \sum\limits_{\nu=1}^{r}a_{\nu,[a,b]}\otimes_{j=1}^{n}{\exp}\left( -\alpha_{\nu,[a,b]} \left( M^{(j)}\right)^{-1}A^{(j)}\right)\right) \left( \otimes_{j=1}^{n}\left( M^{(j)}\right)^{-1}\right), $$
where the coefficients aν,αν > 0 are such that
$$ \begin{array}{@{}rcl@{}} \varepsilon\left( \frac{1}{x},[a,b],r\right) & :=& \|\frac{1}{x}-\sum\limits_{\nu=1}^{r}a_{\nu,[a,b]}\mathrm{e}^{-\alpha_{\nu,[a,b]}x}\|_{[a,b]}\\ & =& \inf\left\{\|\frac{1}{x}-\sum\limits_{\nu=1}^{r}b_{\nu}\mathrm{e}^{-\beta_{\nu}x}\|_{[a,b],\infty}: ~b_{\nu},\beta_{\nu}\in\mathbb{R}\right\} \end{array} $$
with \(a:={\sum }_{j=1}^{n}\lambda _{\min \limits }^{(j)}\) and \(b:={\sum }_{j=1}^{n}{\lambda }_{\max \limits }^{(j)}\). The error can be estimated by
$$ \|A^{-1}-B\|_{2}\leq\varepsilon\left( \frac{1}{x},[a,b],r\right)\|M^{-1}\|_{2}, $$
where \(M=\otimes _{j=1}^{n}M^{(j)}\).
Theorem 5 shows how the inverse of matrices of the form (19) can be approximated by sums of matrix exponentials. It is based on the approximability of the function 1/x by sums of exponentials in the interval [a,b]. We refer to [14, 16] for details how to choose r and the coefficients aν,[a,b], αν,[a,b] in order to reach a given error tolerance \(\varepsilon \left (\frac {1}{x},[a,b],r\right )\). Note that the interval [a,b] where 1/x needs to be approximated depends on the matrices A(j) and M(j). Thus, if A changes a and b need to be recomputed which in turn has an influence on the optimal choice of the parameters aν,[a,b] and αν,[a,b].
Numerical methods based on Theorem 5 can only be efficient if the occurring matrix exponential can be evaluated at low cost. In our setting we will need to compute the matrices \(\exp \big (-\alpha _{\nu ,[a,b]}{M}_{1}^{-1}A_{1}\big )\) and \(\exp \big (-\alpha _{\nu ,[a,b]}(M^{\prime })^{-1}A^{\prime }\big )\). The evaluation of the first matrix will typically be simpler. In the case where a finite difference scheme is employed and A1 is a tridiagonal Toeplitz matrix while M1 is the identity, the matrix exponential can be computed by diagonalizing A1, e.g., A1 = SD1S− 1, and using \(\exp \big (-\alpha _{\nu ,[a,b]}{M}_{1}^{-1}A_{1}\big )=S\exp \big (-\alpha _{\nu ,[a,b]}D_{1}\big )S^{-1}\). The computation of exponentials for general matrices is more involved. We refer to [22] for an overview of different numerical methods. Here, we will make use of the Dunford–Cauchy integral (see [15]). For a matrix \(\tilde {M}\) we can write
$$ {\exp}(-\tilde{M}) = \frac{1}{2\pi\mathrm{i}}\oint_{\mathcal{C}}(\zeta I-\tilde{M})^{-1}\mathrm{e}^{-\zeta}d\zeta $$
for a contour \(\mathcal {C}=\partial D\) which encircles all eigenvalues of \(\tilde {M}\). We assume here that \(\tilde {M}\) is positive definite. Then the spectrum of \(\tilde {M}\) satisfies \(\sigma (\tilde {M})\subset (0,\|M\|]\) and the following (infinite) parabola
$$ \left\{\zeta(s)=x(s)+\mathrm{i}y(s):x(s):=s^{2},~y(s):=-s\quad\text{for }s\in\mathbb{R}\right\} $$
can be used as integration curve \(\mathcal {C}\). The substitution \(\zeta \rightarrow s^{2}-\mathrm {i}s\) then leads to
$$ {\exp}(-\tilde{M}) = {\int}_{-\infty}^{\infty}\underbrace{\left( \frac{1}{2\pi\mathrm{i}}(s^{2}-\mathrm{i}s)I-\tilde{M}\right)^{-1}\mathrm{e}^{-s^{2}+\mathrm{i}s}(2s-\mathrm{i})}_{=:G(s)}ds. $$
(20)
The integrand decays exponentially for \(s\rightarrow \pm \infty \). Therefore (20) can be efficiently approximated by sinc quadrature, i.e.,
$$ {\exp}(-\tilde{M}) = {\int}_{-\infty}^{\infty}G(s)ds\approx\mathfrak{h}\sum\limits_{\nu=-N}^{N}G(\nu\mathfrak{h}), $$
(21)
where \(\mathfrak {h}>0\) and should be chosen s.t. \(\mathfrak {h}=\mathcal {O}\left ((N+1)^{-2/3}\right )\). We refer to [15] for an introduction to sinc quadrature and for error estimates for the approximation in (21). The parameters \(\mathfrak {h}\) and N in our implementation have been chosen such that quadrature errors become negligible compared to the overall discretisation error. For practical computations, the halving rule (see [15, §14.2.2.2]) could be faster while the Dunford–Schwartz representation with sinc quadrature is more suited for an error analysis.