We study the approximation properties of a wide class of finite element differential forms on curvilinear cubic meshes in n dimensions. Specifically, we consider meshes in which each element is the image of a cubical reference element under a diffeomorphism, and finite element spaces in which the shape functions and degrees of freedom are obtained from the reference element by pullback of differential forms. In the case where the diffeomorphisms from the reference element are all affine, i.e., mesh consists of parallelotopes, it is standard that the rate of convergence in L 2 exceeds by one the degree of the largest full polynomial space contained in the reference space of shape functions. When the diffeomorphism is multilinear, the rate of convergence for the same space of reference shape function may degrade severely, the more so when the form degree is larger. The main result of the paper gives a sufficient condition on the reference shape functions to obtain a given rate of convergence. Mathematics Subject Classification (2010)

The present work concerns the approximation of the solution map S associated to the parametric Helmholtz boundary value problem, i.e., the map which associates to each (real) wavenumber belonging to a given interval of interest the corresponding solution of the Helmholtz equation. We introduce a least squares rational Padé-type approximation technique applicable to any meromorphic Hilbert space-valued univariate map, and we prove the uniform convergence of the Padé approximation error on any compact subset of the interval of interest that excludes any pole. This general result is then applied to the Helmholtz solution map S, which is proven to be meromorphic in ℂ, with a pole of order one in every (single or multiple) eigenvalue of the Laplace operator with the considered boundary conditions. Numerical tests are provided that confirm the theoretical upper bound on the Padé approximation error for the Helmholtz solution map.

Abstract. We study the single-phase flow in a saturated, bounded heterogeneous porous medium. We model the permeability as a log-normal random field. We perform a perturbation analysis, expanding the solution in Taylor series. The approximation properties of the Taylor polynomial are studied, and the local convergence of the Taylor series is proved. With a counterexample we show that, in general, the Taylor series is not globally convergent to the stochastic solution as the polynomial degree goes to infinity. Nevertheless, for small variability of the permeability field and low degree of the Taylor polynomial, the perturbation approach is feasible and provides a good approximation of both the stochastic solution and the statistical moments of the stochastic solution. We derive an upper bound on the norm of the residual of the Taylor series, which predicts the optimal degree of the Taylor polynomial to consider. The upper bound is quite pessimistic. In the simple case of a permeability field described by only one random variable, we show numerically that a simple "tuning" of the upper bound, which uses estimates of the growth of the derivatives, provides sharp bounds.

In this work, we consider the approximation of Hilbert space-valued meromorphic functions that arise as solution maps of parametric PDEs whose operator is the shift of an operator with normal and compact resolvent, e.g. the Helmholtz equation. In this restrictive setting, we propose a simplified version of the Least-Squares Padé approximation technique introduced in [6]. In particular, the estimation of the poles of the target function reduces to a low-dimensional eigenproblem for a Gramian matrix, allowing for a robust and efficient numerical implementation (hence the "fast" in the name). Moreover, we prove several theoretical results that improve and extend those in [6], including the exponential decay of the error in the approximation of the poles, and the convergence in measure of the approximant to the target function. The latter result extends the classical one for scalar Padé approximation to our functional framework. We provide numerical results that confirm the improved accuracy of the proposed method with respect to the one introduced in [6] for differential operators with normal and compact resolvent.

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