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We consider the map
$\mathcal{S}:\mathbb{C}\to H^1_0(\Omega)=\{v\in H^1(D), v|_{\partial\Omega}=0\}$, which associates a complex value
*z* with the weak solution of the (complex-valued) Helmholtz problem
$-\Delta u-zu=f$ over
$\Omega$ for some fixed
$f\in L^2(\Omega)$. We show that
$\mathcal{S}$ is well-defined and meromorphic in
$\mathbb{C}\setminus\Lambda$,
$\Lambda=\{\lambda_\alpha\}_{\alpha=1}^\infty$ being the (countable, unbounded) set of (real, non-negative) eigenvalues of the Laplace
operator (restricted to
$H^1_0(\Omega)$). In particular, it holds
$\mathcal{S}(z)=\sum_{\alpha=1}^\infty\frac{s_\alpha}{\lambda_\alpha-z}$, where the elements of
$\{s_\alpha\}_{\alpha=1}^\infty\subset H^1_0(\Omega)$ are pair-wise orthogonal with respect to the
$H^1_0(\Omega)$ inner product. We define a Pad\'e-type approximant of any map as above around
$z_0\in\mathbb{C}$: given some integer degrees of the numerator and denominator respectively,
$M,N\in\mathbb{N}$, the exact map is approximated by a rational map
$\mathcal{S}_{[M/N]}:\mathbb{C}\setminus\Lambda\to H^1_0(\Omega)$. We define such approximant within a Least-Squares framework, through the minimization
of a suitable functional based on samples of the target solution map and of its derivatives
at
$z_0$. In particular, the denominator of the approximant is the minimizer (under some normalization
constraints) of the
$H^1_0(\Omega)$ norm of a Taylor coefficient of
$Q\mathcal{S}$, as
*Q* varies in the space of polynomials with degree
$\leq N$. The numerator is then computed by matching as many terms as possible of the Taylor
series of
$\mathcal{S}$ with those of
$\mathcal{S}_{[M/N]}$, analogously to the classical Pad\'e approach. The resulting approximant is shown
to converge, as
$M+N$ goes to infinity, to the exact map
$\mathcal{S}_{[M/N]}$ in the
$H^1_0(\Omega)$ norm for values of the parameter sufficiently close to
$z_0$ (a sharp bound on the region of convergence is given). Moreover, it is proven that
the approximate poles converge exponentially (as
*M* goes to infinity) to the
*N* elements of
$\Lambda$ closer to
$z_0$.

ScienceOpen

10.14293/P2199-8442.1.SOP-MATH.HWEDJF.v1

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