118

views

0

recommends

- Record: found
- Abstract: found
- Poster: found

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

This work has been published open access under Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com.