Mixed Finite Element Methods and Applications

[no abstract available

Superconvergence of the effective Cauchy stress in computational homogenization of inelastic materials

We provide theoretical investigations and empirical evidence that the effective stresses in computational homogenization of inelastic materials converge with a higher rate than the local solution fields. Due to the complexity of industrial-scale microstructures, computational homogenization methods often utilize a rather crude approximation of the microstructure, favoring regular grids over accurate boundary representations. As the accuracy of such an approach has been under continuous verification for decades, it appears astonishing that this strategy is successful in homogenization, but is seldom used on component scale. A part of the puzzle has been solved recently, as it was shown that the effective elastic properties converge with twice the rate of the local strain and stress fields. Thus, although the local mechanical fields may be inaccurate, the averaging process leads to a cancellation of errors and improves the accuracy of the effective properties significantly. Unfortunately, the original argument is based on energetic considerations. The straightforward extension to the inelastic setting provides superconvergence of (pseudoelastic) potentials, but does not cover the primary quantity of interest: the effective stress tensor. The purpose of the work at hand is twofold. On the one hand, we provide extensive numerical experiments on the convergence rate of local and effective quantities for computational homogenization methods based on the fast Fourier transform. These indicate the superconvergence effect to be valid for effective stresses, as well. Moreover, we provide theoretical justification for such a superconvergence based on an argument that avoids energetic reasoning

Ws,pW^{s,p}-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray-Lions problems

In this work we prove optimal $W^{s,p}$-approximation estimates (with $p\in[1,+\infty]$) for elliptic projectors on local polynomial spaces. The proof hinges on the classical Dupont--Scott approximation theory together with two novel abstract lemmas: An approximation result for bounded projectors, and an $L^p$-boundedness result for $L^2$-orthogonal projectors on polynomial subspaces. The $W^{s,p}$-approximation results have general applicability to (standard or polytopal) numerical methods based on local polynomial spaces. As an illustration, we use these $W^{s,p}$-estimates to derive novel error estimates for a Hybrid High-Order discretization of Leray--Lions elliptic problems whose weak formulation is classically set in $W^{1,p}(\Omega)$ for some $p\in(1,+\infty)$. This kind of problems appears, e.g., in the modelling of glacier motion, of incompressible turbulent flows, and in airfoil design. Denoting by $h$ the meshsize, we prove that the approximation error measured in a $W^{1,p}$-like discrete norm scales as $h^{\frac{k+1}{p-1}}$ when $p\ge 2$ and as $h^{(k+1)(p-1)}$ when $p<2$.Comment: keywords: $W^{s,p}$-approximation properties of elliptic projector on polynomials, Hybrid High-Order methods, nonlinear elliptic equations, $p$-Laplacian, error estimate

Analysis of a finite volume element method for the Stokes problem

In this paper we propose a stabilized conforming finite volume element method for the Stokes equations. On stating the convergence of the method, optimal a priori error estimates in different norms are obtained by establishing the adequate connection between the finite volume and stabilized finite element formulations. A superconvergence result is also derived by using a postprocessing projection method. In particular, the stabilization of the continuous lowest equal order pair finite volume element discretization is achieved by enriching the velocity space with local functions that do not necessarily vanish on the element boundaries. Finally, some numerical experiments that confirm the predicted behavior of the method are provide

Accurate gradient computations at interfaces using finite element methods

New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is not only to get an accurate solution but also an accurate first order derivative at the interface (from each side). The key in 1D is to use the idea from \cite{wheeler1974galerkin}. For 2D interface problems, the idea is to introduce a small tube near the interface and introduce the gradient as part of unknowns, which is similar to a mixed finite element method, except only at the interface. Thus the computational cost is just slightly higher than the standard finite element method. We present rigorous one dimensional analysis, which show second order convergence order for both of the solution and the gradient in 1D. For two dimensional problems, we present numerical results and observe second order convergence for the solution, and super-convergence for the gradient at the interface