Continuity of approximation by least-squares multivariate Padé approximants

We prove that if (uh(z))h>0 is a family of meromorphic functions which converges to a meromorphic function u(z), then [M,N]uh→u when (h,M)→(0,+∞), where [M,N]uh denotes the least-squares multivariate Padé approximants (LSPA) of uh. This property is fundamental when using the LSPA to approximate the solution of a partial differential equations problem depending on some parameters. We illustrate it on a structural mechanics eigenproblem with variable damping coefficient.