Abstract. An enriched nonconforming Multiscale Finite Element Method (MsFEM) to solve viscous incompressible flow problems in genuine heterogeneous or porous media was proposed in [Q. Feng, G. Allaire, and P. Omnes, Multiscale Model. Simul., 20 (2022), pp. 462–492]. The main feature of this MsFEM is the consideration of high-order sets of weighting functions: for the velocity, they are polynomials of order [Formula: see text] on the faces and of order [Formula: see text] in the volume of the elements; for the pressure they are polynomials of order [Formula: see text] in the element volume. In the previously cited reference, only the case [Formula: see text] was numerically tested. The present paper proposes the first implementation for the case [Formula: see text] in two and three dimensions. Furthermore, a discrete analysis of this MsFEM applied to the Stokes problem in heterogeneous media is performed. In particular, a first error estimate is obtained, proving the convergence of this MsFEM for the Stokes problem in periodic perforated media. In addition, it has been shown in the previously cited reference that the continuous local problems involved in this MsFEM are well-posed. Here, their discrete counterparts are also proved to be well-posed, for any [Formula: see text] in two dimensions and for [Formula: see text] equal to 1 and 2 in three dimensions, with a judicious choice of nonconforming pairs of finite elements.

We introduce a family of scalar non-conforming finite elements with second- and third-order accuracy with respect to the -norm on tetrahedra. Their vector-valued versions generate, together with discontinuous pressure approximations of order one and two, respectively, inf-sup stable finite element pairs with convergence order two and three for the Stokes problem in energy norm.

This paper addresses the homogenization of the Stokes equations in a periodic perforated domain. The homogenized model is known to correspond to Darcy's law in the full domain. We established a sharp convergence rate O(root epsilon) for the energy norm of the difference in velocities, where epsilon represents the size of the solid obstacles. This was achieved by using a two-scale asymptotic expansion of the Stokes equations and a new construction of a cutoff function that avoids the introduction of boundary layers. The main novelty is that our analysis applies to the physically relevant case of a porous medium where each fluid and solid part is a connected subdomain.

We propose and analyse the optimized Schwarz waveform relaxation (OSWR) method for the unsteady incompressible Stokes equations. Well-posedness of the local subdomain problems with Robin boundary conditions is proved. Convergence of the velocity is shown through energy estimates; however, pressure converges only up to constant values in the subdomains, and an astute correction technique is proposed to recover these constants from the velocity. The convergence factor of the OSWR algorithm is obtained through a Fourier analysis, and allows to efficiently optimize the space-time Robin transmission conditions involved in the OSWR method. Then, numerical illustrations for the two-dimensional unsteady incompressible Stokes system are presented to illustrate the performance of the OSWR algorithm.

This article focuses on discretizing the convection–diffusion–reaction equation (concentration or heat), which is time-dependent and coupled with the Darcy equation. The Darcy system is discretized using the Raviart–Thomas finite element, while the concentration equation is discretized using the Lagrange finite element of order one in space and the Euler method in time. We present optimal a posteriori error estimates utilizing two computable error indicators: linearization and discretization indicators. Ultimately, numerical experiments demonstrate and confirm the efficacy of the proposed method, and show comparisons with previous works.

Our aim is to develop a robust and flexible code to simulate flows in nuclear core reactors. Discontinuous Galerkin schemes are thus proposed here to solve the Stokes problem, which stands as a fundamental element in fluid mechanics. A priori error estimates are provided when the solution is weakly regular. Details on how to solve the large-scale linear system are given. We moreover explain how to discretize a singular source term. Finally, we give numerical results of two test-cases for which the solution is weakly regular.

Error control by means of a posteriori error estimators or indicators and adaptive discretizations, such as adaptive mesh refinement, have emerged in the late seventies.Since then, numerous theoretical developments and improvements have been made, as well as the first attempts to introduce them into real-life industrial applications.The present introductory chapter provides an overview of the subject, highlights some of the achievements to date and discusses possible perspectives.

Fused Filament Fabrication (FFF) is a promising technology that is largely developed in small series, as this technology optimizes supply chains by reducing production time and costs. However, its shortcomings have slowed its adoption as a dominant production technology. Among its weaknesses, this work focuses on geometric and dimensional accuracy within tolerance range. There is a need for understanding the sources of geometrical inaccuracies and for methods of characterizing them, in order to modify the input parameters to eventually obtain the desired geometry. This work first focuses on the geometric and dimensional accuracy of parts printed by the FFF process by studying the influence of the inner radius of a cylindrical part, the type of material and the type of filling pattern. The levels with the greatest dimensional dispersion are the largest radius, the nylon material, and the hexagonal filling pattern. Secondly, a defect characterization method associated with a parametric mathematical model is developed. The 3D scanner enables the retrieval of the coordinates of the printed geometry; this allows to characterize the errors with respect to the theoretical 3D model and to modelize the printed part by a series of ellipses of which we obtain the analytical equations, as a first step of a correction process.

We consider the lowest order - so called B-grid - discretization on triangular cells which is common in the ocean or atmosphere simulation science, with pressure field at the vertices of the triangles and the velocity field at the cell centers. It is known that this discretization presents spurious modes due to a too large number of velocities unknowns and that these modes should be damped; since preservation of approximate geostrophic equilibrium (GE) may be of primary importance in the simulations, we propose and analyze damping operators that, when applied to the linearized constant coefficient configuration, exactly preserve discrete versions of the GEs. We also analyze their effects in more general configurations and their possible coupling with more standard damping operators. These strategies cover triangular variants of the Low Froude (LF) scheme [1], the Apparent Topography (AT) scheme [2], [3], as well as a new modification of it. The LF scheme has no numerical diffusion terms in the pressure equation and damps the normal velocity jumps between adjacent cells in the momentum equation. In the AT scheme, numerical diffusion is added in the pressure equation in such a way that it does not impact the discrete GE. However, since the AT scheme alone cannot be proved to be damping through energy estimates and does not preserve the space which is orthogonal to the discrete GE, we also study what we call the Modified Apparent Topography (MAT) that satisfies these important properties. We also suggest semi-implicit time discretizations that preserve these properties. Extensions to the full non-linear equations are also provided. Numerical experiments illustrate the performance of the new schemes.

We study the two dimensional time dependent Large Eddy Simulation method applied to the incompressible Navier–Stokes system with Smagorinsky’s eddy viscosity model and a filter width that depends on the local mesh size. The discrete model is based on the implicit Euler scheme and a conforming finite element method for the time and space discretizations, respectively. We establish a reliable and efficient a posteriori error estimation between the numerical LES solution and the exact solution of the original Navier–Stokes system, which involves three types of error indicators respectively related to the filter and to the discretizations in time and space. Numerical results show the effectiveness of adaptive simulations

In this paper we study the a posteriori error estimates for the time dependent Navier-Stokes system coupled with the convection-diffusion-reaction equation. The problem is discretized in time using the implicit Euler method and in space using the finite element method. We establish a posteriori error estimates with two types of computable error indicators, the first one linked to the space discretization and the second one to the time discretization. Finally, numerical investigations are performed and presented.

We propose a new approach that provides new results in the convergence analysis of optimized Schwarz waveform relaxation (OSWR) iterations for parabolic problems, and allows to define efficient optimized Robin parameters that depend on the targeted iteration count, a property that is shared by the actual observed optimal parameters, while traditional Fourier analysis in the time direction leads to iteration independent parameters. This new approach is based on the exact resolution of the time semi-discrete error equations. It allows to recommend a couple (number of iterations, Robin parameter) to reach a given accuracy. While the general ideas may apply to an arbitrary space dimension, the analysis is first presented in the one dimensional case. Numerical experiments illustrate the performance obtained with such iteration-dependent optimized Robin parameters.

. We propose a new approach to analyze the convergence of optimized Schwarz wave-form relaxation (OSWR) iterations for parabolic problems. Departing from traditional Fourier analysis in the time direction, we explicitly solve the equations obtained using the backward Euler scheme in time, and deduce convergence properties from this solution, in the two subdomains case. Convergence is proven for any positive Robin parameter (or couple of such parameters in the two-sided case). We also show that, for any fixed value of the number of time steps, the convergence depends on a single parameter which is a combination of the diffusion coefficient, the time step and the Robin parameter. A convergence result in a finite number of iterations is also proven for a well-chosen value of the Robin parameters. This approach allows us to define efficient optimized Robin parameters that depend on the number of iterations one wishes to perform, and to recommend a couple (number of iterations, Robin parameter) to reach a given accuracy. Numerical experiments illustrate the performance of such iteration-dependent optimized Robin parameters, compared to the observed optimal ones (which also depend on the number of iterations performed). A comparison is also given with optimized parameters derived by classical Fourier transform analysis on the continuous problem (which are independent of the iterations).

This paper addresses an enriched nonconforming multiscale finite element method (MsFEM) to solve viscous incompressible flow problems in genuine heterogeneous or porous media. In the work of [B. P. Muljadi, et al., Multiscale Model. Simul., 13 (2015), pp. 1146--1172] and [G. Jankowiak and A. Lozinski, arXiv:1802.04389, 2018], a nonconforming MsFEM has been first developed for Stokes problems in such media. Based on these works, we propose an innovative enriched nonconforming MsFEM where the approximation space of both velocity and pressure are enriched by weighting functions which are defined by polynomials of higher-degree. Numerical experiments show that this enriched nonconforming MsFEM improves significantly the accuracy of the nonconforming MsFEMs. Theoretically, this method provides a general framework which allows one to find a good compromise between the accuracy of the method and the computing costs, by varying the degrees of polynomials.

We propose and analyze a new parallel paradigm that uses both the time and the space directions. The original approach couples the Parareal algorithm with incomplete optimized Schwarz waveform relaxation (OSWR) iterations. The analysis of this coupled method is presented for a one-dimensional advection-reaction-diffusion equation. We also prove a general convergence result for this method via energy estimates. Numerical results for two-dimensional advection-diffusion problems and for a diffusion equation with strong heterogeneities are presented to illustrate the performance of the coupled Parareal-OSWR algorithm.

In this paper, we study in two and three space dimensions, the a posteriori error estimates for the large eddy simulation (LES) applied to the Navier–Stokes system. We begin by introducing the Navier–Stokes and the corresponding LES equations. Then we introduce the corresponding discrete problem based on the finite element method. We establish a posteriori error estimation with three types of error indicators related to the filter of the LES method, to the discretization and to the linearization. Finally, numerical investigations are shown and discussed.

Classical finite volume schemes for the Euler system are not accurate at low Mach number and some fixes have to be used and were developed in a vast literature over the last two decades. The question we are interested in in this article is: What about if the porosity is no longer uniform? We first show that this problem may be understood on the linear wave equation taking into account porosity. We explain the influence of the cell geometry on the accuracy property at low Mach number. In the triangular case, the stationary space of the Godunov scheme approaches well enough the continuous space of constant pressure and divergence-free velocity, while this is not the case in the Cartesian case. On Cartesian meshes, a fix is proposed and accuracy at low Mach number is proved to be recovered. Based on the linear study, a numerical scheme and a low Mach fix for the non-linear system, with a non-conservative source term due to the porosity variations, is proposed and tested.

In the aim to find the simplest and most efficient shape of a noise absorbing wall to dissipate the acoustical energy of a sound wave, we consider a frequency model described by the Helmholtz equation with a damping on the boundary. The well-posedness of the model is shown in a class of domains with d-set boundaries (N -1 <= d < N). We introduce a class of admissible Lipschitz boundaries, in which an optimal shape of the wall exists in the following sense: We prove the existence of a Radon measure on this shape, greater than or equal to the usual Lebesgue measure, for which the corresponding solution of the Helmholtz problem realizes the infimum of the acoustic energy defined with the Lebesgue measure on the boundary. If this Radon measure coincides with the Lebesgue measure, the corresponding solution realizes the minimum of the energy. For a fixed porous material, considered as an acoustic absorbent, we derive the damping parameters of its boundary from the corresponding time-dependent problem described by the damped wave equation (damping in volume).

In this article, we study the time dependent convection–diffusion–reaction equation coupled with the Darcy equation. We propose and analyze two numerical schemes based on finite element methods for the discretization in space and the implicit Euler method for the discretization in time. An optimal a priori error estimate is then derived for each numerical scheme. Finally, we present some numerical experiments that confirm the theoretical accuracy of the discretization.