Alexandre L. Madureira

CONVERGENCE ANALYSIS OF A MULTISCALE FINITE ELEMENT METHOD FOR SINGULARLY PERTURBED PROBLEMS

In this paper we perform an error analysis for a multiscale finite element method for singularly perturbed reaction-diffusion equations. Such a method is based on enriching the usual piecewise linear finite element trial spaces with local solutions of the original problem, but the approach does not require these functions to vanish on each element edge. Piecewise linear plus bubbles are the choice for the test functions allowing static condensation; thus our method is of Petrov--Galerkin type. We perform convergence analysis in different asymptotic regimes, and we show convergence in an appropriate norm with respect to the small parameter. Numerical results show that the new method is able to compute solutions even on coarse meshes.

On the range of applicability of the Reissner-Mindlin and Kirchhoff-Love plate bending models

We show that the Reissner–Mindlin plate bending model has a wider range of applicability than the Kirchhoff–Love model for the approximation of clamped linearly elastic plates. Under the assumption that the body force density is constant in the transverse direction, the Reissner–Mindlin model solution converges to the three-dimensional linear elasticity solution in the relative energy norm for the full range of surface loads. However, for loads with a significant transverse shear effect, the Kirchhoff–Love model fails.

ASYMPTOTICS OF THE POISSON PROBLEM IN DOMAINS WITH CURVED ROUGH BOUNDARIES

Effective boundary conditions (wall laws) are commonly employed to approximate PDEs in domains with rough boundaries, but it is neither easy to design such laws nor to estimate the related approximation error. A two-scale asymptotic expansion based on a domain decomposition result is used here to mitigate such difficulties, and as an application we consider the Poisson equation. The proposed scheme considers rough curved boundaries and allows a complete asymptotic expansion for the solution, highlighting the influence of the boundary curvature. The derivation and estimation of high order effective conditions is a corollary of such development. Sharp estimates for first and second order wall law approximations are considered for different Sobolev norms and show superior convergence rates in the interior of the domain. A numerical test illustrates several of the results obtained here.

Element diameter free stability parameters for stabilized methods applied to fluids

Abstract Stability parameters for stabilized methods in fluids are suggested. The computation of the largest eigenvalue of a generalized eigenvalue problem replaces controversial definitions of element diameters and inverse estimate constants, used heretofore to compute these stability parameters. The design is employed in the advective-diffusive model, incompressible Navier-Stokes equations and the Stokes problem.

A multiscale finite element method for partial differential equations posed in domains with rough boundaries

We propose and analyze a finite element scheme of multiscale type to deal with elliptic partial differential equations posed in domains with rough boundaries. There is no need to assume that the boundary is periodic in any sense, so the method is quite general. On the other hand, if the boundary is periodic we prove convergence of the scheme.

A NEW INTERIOR PENALTY DISCONTINUOUS GALERKIN METHOD FOR THE REISSNER–MINDLIN MODEL

We introduce an interior penalty discontinuous Galerkin finite element method for the Reissner–Mindlin plate model that, as the plates half-thickness ϵ tends to zero, recovers a hp interior penalty discontinuous Galerkin finite element methods for biharmonic equation. Our method does not introduce shear as an extra unknown, and does not need reduced integration techniques. We develop the a priori error analysis of these methods and prove error bounds that are optimal in h and uniform in ϵ. Numerical tests, that confirm our predictions, are provided.

ASYMPTOTIC ESTIMATES OF HIERARCHICAL MODELING

In this paper we propose a way to analyze certain classes of dimension reduction models for elliptic problems in thin domains. We develop asymptotic expansions for the exact and model solutions, having the thickness as small parameter. The modeling error is then estimated by comparing the respective expansions, and the upper bounds obtained make clear the inuence of the order of the model and the thickness on the convergence rates. The techniques developed here allows for estimates in several norms and semi-norms, and also interior estimates (which disregards boundary layers).

Analysis of curvature influence on effective boundary conditions

Abstract The aim of this work is the construction of effective boundary conditions (wall laws) for elliptic problems defined in domains with curved, rough boundaries with periodic wrinkles. We present error estimates for first and second order approximations, and a numerical test. To cite this article: A. Madureira, F. Valentin, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 499–504.

HIERARCHICAL MODELING BASED ON MIXED PRINCIPLES: ASYMPTOTIC ERROR ESTIMATES

We analyze approximation properties of dimension reduction models that are based on mixed principles. The problems of interest are elliptic PDEs in thin domains. The goal is to obtain estimates that take into account both the thickness of the domain and the order of the model. The techniques involved do not require the models to be energy minimizers, and are based on asymptotic expansions for the exact and model solutions. We obtain estimates in several norms and semi-norms, and also interior estimates (which disregards boundary layers).

Modeling PDEs in Domains with Rough Boundaries

We discuss here PDEs defined in domains where at least part of the boundary is rugous. The fully discretization of such domains can be very expensive, and we show two ways to decrease such burden. When the wrinkles are periodic, one way is to avoid the expensive discretization of the rough domain altogether, replacing the rough domain by a smooth one and changing the boundary conditions in such a way that the geometry of the wrinkles is captured. For a general domain, a possibility is to use a domain decomposition approach, solving local problems in parallel in the spirit of the Multiscale Finite Element Method. Asymptotic expansions play a key role in both alternatives, motivating the development of models, and helping in deriving error estimates.