A Staggered Cell-centered Finite Element Method with Grad-div Stabilization for the Stokes Problems

Advanced Topics in Computational Partial Differential Equations: Numerical Methods and Diffpack Programming

In this paper we study the Brinkman model as a unified framework to allow the transition between the Darcy and the Stokes problems. We propose an unconditionally stable low-order finite element approach, which is robust with respect to the whole range of physical parameters, and is based on the combination of stabilized equal-order finite elements with a non-symmetric penalty-free Nitsche method for the weak imposition of essential boundary conditions. In particular, we study the properties of the penalty-free Nitsche formulation for the Brinkman setting, extending a recently reported analysis for the case of incompressible elasticity (Boiveau and Burman, IMA J. Numer. Anal. 36 (2016) 770-795). Focusing on the two-dimensional case, we obtain optimal a priori error estimates in a mesh-dependent norm, which, converging to natural norms in the cases of Stokes or Darcy ows, allows to extend the results also to these limits. Moreover, we show that, in order to obtain robust estimates also in the Darcy limit, the formulation shall be equipped with a Grad-Div stabilization and an additional stabilization to control the discontinuities of the normal velocity along the boundary. The conclusions of the analysis are supported by numerical simulations.

Efficient coupling of finite elements and boundary elements---adaptive procedures and preconditioners

New technologies in computer science applied to numerical computations open the door to alternative approaches to mechanical problems using the finite element method. In classical approaches, theoretical developments often become cumbersome and the computer model which follows shows resemblance with the initial problem statement. The first step in the development consists usually in the analysis of the physics of the problem to simulate. The problem is generally described by a set of equations including partial differential equations. This first model is then replaced by successive equivalent or approximated models. The final result consists in a mathematical description of elemental matrices and algorithms describing the matrix form of the problem. The traditional approach consists then in constructing a computer model, generally complex and often quite different from the original mathematical description, thus making further corrections difficult. Therefore, the crucial problem of both the software architecture and the choice of the appropriate programming language is raised. Partially breaking with this approach, we propose a new approach to develop and program finite element formulations. The approach is based on a hybrid symbolic/numerical approach on the one hand, and on a high level software tool, object-oriented programming (supported here by the languages Smalltalk and C++) on the other hand. The aim of this work is to develop an appropriate environment for the algebraic manipulations needed for a finite element formulation applied to an initial boundary value problem, and also to perform efficient numerical computations. The new environment should make it possible to manage al1 the concepts necessary to solve a physical problem: manipulation of partial differential equations, variational formulations, integration by parts, weak forms, finite element approximations… The concepts manipulated therefore remain closely related to the original mathematical framework. The result of these symbolic manipulations is a set of elemental data (mass matrix, stiffness matrix, tangent stiffness matrix,…) to be introduced in a classical numerical code. The object-oriented paradigm is essential to the success of the implementation. In the context of the finite element codes, the object-oriented approach has already proved its capacity to represent and handle complex structures and phenomena. This is confirmed here with the symbolic environment for derivation of finite element formulations in which objects such as expression, integral and variational formulation appear. The link between both the numerical world and the symbolic world is based on an object-oriented concept for automatic programmation of matrix forms derived from the finite element method. As a result, a global environment in which the numerical is capable of evolving, using a language close to the natural mathematical one, is achieved. The potential of the approach is further demonstrated, on the one hand, by the wide range of problems solved in linear mechanics (electrodynamics in 1 and 2D, heat diffusion,…) as well as in nonlinear mechanics (advection dominated 1D problem, Navier Stokes problem), and, on the other hand by the diversity of the formulations manipulated (Galerkin formulations, space-time Galerkin formulations continuous in space and discontinuous in time, generalized Galerkin least-squares formulations).

A comparison of the meshless RBF collocation method with finite element and boundary element methods in neutron diffusion calculations

<abstract><p>We investigate a fully discrete modular grad-div (MGD) stabilization algorithm for solving the incompressible micropolar Navier-Stokes equations (IMNSE) model, which couples the incompressible Navier-Stokes equations and the angular momentum equation together. The mixed finite element (FE) method is applied for the spatial discretization. The time discretization is based on the BDF2 implicit scheme for the linear terms and the two-step linearly extrapolated scheme for the convective terms. The considered algorithm constitutes two steps, which involve a post-processing step for linear velocity. First, we decouple the fully coupled IMNSE model into two smaller sub-physics problems at each time step (one is for the linear velocity and pressure, the other is for the angular velocity), which reduces the size of the linear systems to be solved and allows for parallel computing of the two sub-physics problems. Then, in the post-processing step, we only need to solve a symmetrical positive determined grad-div system of linear velocity at each time step, which does not increase the computational complexity by much. However, the post-processing step can improve the solution quality of linear velocity. Moreover, we obtain unconditional stability, and error estimates of the linear velocity and angular velocity. Finally, several numerical experiments involving three-dimensional and two-dimensional settings are used to validate the theoretical findings and demonstrate the benefits of the modular grad-div (MGD) stabilization algorithm.</p></abstract>

The fluid-structure interaction technique provides a paradigm for solving scattering from elastic targets embedded in a fluid by a combination of finite and boundary element methods. In this technique, the finite element method is used to compute the target’s elastic response and the boundary element method with the appropriate Green’s function is used to compute the field in the exterior medium. The two methods are coupled at the surface of the target by imposing the continuity of pressure and normal displacement. This results in a self-consistent boundary element method that can be used to compute the scattered field anywhere in the surrounding environment. The method reduces a finite element problem to a boundary element one with drastic reduction in the number of unknowns, which translates to a significant reduction in numerical cost. In this talk, the method is extended to compute scattering from multiple targets by self-consistently accounting for all interactions between them. The model allows to identify block matrices responsible for the interaction between targets, which proves useful in many applications. The model is tested by comparing its results with those measured involving two aluminum cylinders one of which is excited by modulated radiation pressure.

Note on the effect of grad-div stabilization on calculating drag and lift coefficients

A two-level finite element method with grad-div stabilizations for the incompressible Navier–Stokes equations

Stress recovery for the particle-in-cell finite element method

A stabilized finite volume element method for a coupled Stokes–Darcy problem

We analyze the spatially semidiscrete piecewise linear finite volume element method for parabolic equations in a convex polygonal domain in the plane. Our approach is based on the properties of the standard finite element Ritz projection and also of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method. Because the domain is polygonal, special attention has to be paid to the limited regularity of the exact solution. We give sufficient conditions in terms of data that yield optimal order error estimates in L2 and H1. The convergence rate in the L∞ norm is suboptimal, the same as in the corresponding finite element method, and almost optimal away from the corners. We also briefly consider the lumped mass modification and the backward Euler fully discrete method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004

Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation

Fluid flow in porous media is a multiscale process where the effective dynamics, which is often the goal of a computation, depends strongly on the porous micro structure. Resolving the micro structure in the whole porous medium can, however, be prohibitive. Novel numerical methods that efficiently approximate the effective flow but resolve only a carefully selected reduced portion of the porous structure are of great interest. In this thesis we propose new numerical multiscale methods for Stokes flow in two- and three-scale porous media. First, we propose the Darcy--Stokes finite element heterogeneous multiscale method (DS-FE-HMM). The method is based on solving the Darcy equation on a macroscopic mesh using the finite element method with numerical quadrature, where the unknown permeability is recovered from micro finite element solutions of Stokes problems that are defined in sampling domains centered at macroscopic quadrature points. An adaptive scheme based on a posteriori error analysis is proposed, where micro-macro mesh refinement is driven by residual-based indicators that quantify both the micro and macro errors. Second, to address the increasing cost of solving the micro problems as the macroscopic mesh is refined, we combine the DS-FE-HMM with reduced basis (RB) method and propose a new multiscale method called the RB-DS-FE-HMM. Efficiency and accuracy of the method relies on a parametrization of the micro geometries and on the Petrov-Galerkin RB formulation that provides a stable and fast evaluation of the effective permeability. A residual-based adaptive mesh refinement scheme is proposed for the macroscopic problem. To achieve a conservative approximation we also combine and analyze a coupling of the RB method with a different macroscopic scheme based on the discontinuous Galerkin finite element method (DG-FEM). Finally, we consider a three-scale porous media model with macro, meso, and micro scale. At the intermediate meso scale the medium is composed of fluid and porous parts and the fluid flow is modeled with the Stokes-Brinkman equation. A three-scale numerical method is derived and an efficient algorithm based on the RB method and empirical interpolation method on the micro and meso scale is proposed.

This thesis considers conforming finite element discretizations for the time-dependent Oberbeck-Boussinesq model with a pressure-correction projection scheme of second order in time. Discrete inf-sup stability of the ansatz spaces for velocity and pressure is assumed. For handling poor mass conservation, a stabilization of the incompressibility constraint, the so called grad-div stabilization, is considered. Furthermore, a local projection stabilization method in streamline direction (LPS SU) is applied for velocity and temperature for dealing with dominating convection. Numerical analysis is performed both with respect to the semi-discretization in space and for the fully discretized model: Stability and convergence results are given and a suitable design of stabilization parameters is proposed. Here, grad-div stabilization proves to be essential for robustness of this approach. These findings are validated by various numerical experiments. Analytical examples are considered to verify convergence rates in space and time. In addition, more realistic isothermal and non-isothermal flow examples are investigated and a suitable parameter choice within the bounds of the theoretical results is obtained experimentally.