Rayleigh-Ritz approximation of the inf-sup constant for the divergence

A numerical scheme for computing approximations to the inf-sup constant of the divergence operator in bounded Lipschitz polytopes in R$^{n}$ is proposed. The method is based on a conforming approximation of the pressure space based on piecewise polynomials of some fixed degree k $\geq \$0. The scheme can be viewed as a Rayleigh–Ritz method and it gives monotonically decreasing approximations of the inf-sup constant under mesh refinement. The new approximation replaces the H⁻¹ norm of a gradient by a discrete H⁻¹ norm which behaves monotonically under mesh refinement. By discretizing the pressure space with piecewise polynomials, upper bounds to the inf-sup constant are obtained. Error estimates are presented that prove convergence rates for the approximation of the inf-sup constant provided it is an isolated eigenvalue of the corresponding non-compact eigenvalue problem; otherwise, plain convergence is achieved. Numerical computations on uniform and adaptive meshes are provided

Two conjectures on the Stokes complex in three dimensions on Freudenthal meshes

In recent years, a great deal of attention has been paid to discretizations of the incompressible Stokes equations that exactly preserve the incompressibility constraint. These are of substantial interest because these discretizations are pressure-robust; i.e., the error estimates for the velocity do not depend on the error in the pressure. Similar considerations arise in nearly incompressible linear elastic solids. Conforming discretizations with this property are now well understood in two dimensions but remain poorly understood in three dimensions. In this work, we state two conjectures on this subject. The first is that the Scott–Vogelius element pair is inf-sup stable on uniform meshes for velocity degree k≥4; the best result available in the literature is for k≥6. The second is that there exists a stable space decomposition of the kernel of the divergence for k≥5. We present numerical evidence supporting our conjectures

A nonconforming pressure-robust finite element method for the Stokes equations on anisotropic meshes

Most classical finite element schemes for the (Navier-)Stokes equations are neither pressure-robust, nor are they inf-sup stable on general anisotropic triangulations. A lack of pressure-robustness may lead to large velocity errors, whenever the Stokes momentum balance is dominated by a strong and complicated pressure gradient. It is a consequence of a method, which does not exactly satisfy the divergence constraint. However, inf-sup stable schemes can often be made pressure-robust just by a recent, modified discretization of the exterior forcing term, using $\mathbf{H}(\operatorname{div})$-conforming velocity reconstruction operators. This approach has so far only been analyzed on shape-regular triangulations. The novelty of the present contribution is that the reconstruction approach for the Crouzeix-Raviart method, which has a stable Fortin operator on arbitrary meshes, is combined with results on the interpolation error on anisotropic elements for reconstruction operators of Raviart-Thomas and Brezzi-Douglas-Marini type, generalizing the method to a large class of anisotropic triangulations. Numerical examples confirm the theoretical results in a 2D and a 3D test case

T-coercivity for solving Stokes problem with nonconforming finite elements

We propose to analyse the discretization of the Stokes problem with nonconforming finite elements in light of the T-coercivity (cf. [1] for Helmholtz-like problems, see [2], [3] and [4] for the neutron diffusion equation). We propose explicit expressions of the stability constants. Finally, we give numerical results illustrating the importance of using divergence-free velocity reconstruction

A nonconforming pressure-robust finite element method for the Stokes equations on anisotropic meshes

Most classical finite element schemes for the (Navier--)Stokes equations are neither pressure-robust, nor are they inf-sup stable on general anisotropic triangulations. A lack of pressure-robustness may lead to large velocity errors, whenever the Stokes momentum balance is dominated by a strong and complicated pressure gradient. It is a consequence of a method, which does not exactly satisfy the divergence constraint. However, inf-sup stable schemes can often be made pressure-robust just by a recent, modified discretization of the exterior forcing term, using H(div)-conforming velocity reconstruction operators. This approach has so far only been analyzed on shape-regular triangulations. The novelty of the present contribution is that the reconstruction approach for the Crouzeix--Raviart method, which has a stable Fortin operator on arbitrary meshes, is combined with results on the interpolation error on anisotropic elements for reconstruction operators of Raviart--Thomas and Brezzi--Douglas--Marini type, generalizing the method to a large class of anisotropic triangulations. Numerical examples confirm the theoretical results in a 2D and a 3D test case

Mathematical analysis and numerical approximation of a general linearized poro-hyperelastic model

Abstract We describe the behavior of a deformable porous material by means of a poro-hyperelastic model that has been previously proposed in Chapelle and Moireau (2014) under general assumptions for mass and momentum balance and isothermal conditions for a two-component mixture of fluid and solid phases. In particular, we address here a linearized version of the model, based on the assumption of small displacements. We consider the mathematical analysis and the numerical approximation of the problem. More precisely, we carry out firstly the well-posedness analysis of the model. Then, we propose a numerical discretization scheme based on finite differences in time and finite elements for the spatial approximation; stability and numerical error estimates are proved. Particular attention is dedicated to the study of the saddle-point structure of the problem, that turns out to be interesting because velocities of the fluid phase and of the solid phase are combined into a single quasi-incompressibility constraint. Our analysis provides guidelines to select the componentwise polynomial degree of approximation of fluid velocity, solid displacement and pressure, to obtain a stable and robust discretization based on Taylor–Hood type finite element spaces. Interestingly, we show how this choice depends on the porosity of the mixture, i.e. the volume fraction of the fluid phase

Computational Engineering

This Workshop treated a variety of finite element methods and applications in computational engineering and expanded their mathematical foundation in engineering analysis. Among the 53 participants were mathematicians and engineers with focus on mixed and nonstandard finite element schemes and their applications