Salim Meddahi

A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. Part II: a posteriori error analysis

This is the second part of a work dealing with a low-order mixed finite element method for a class of nonlinear Stokes models arising in quasi-Newtonian fluids. In the first part we showed that the resulting variational formulation is given by a twofold saddle point operator equation, and that the corresponding Galerkin scheme becomes well posed with piecewise constant functions and Raviart–Thomas spaces of lowest order as the associated finite element subspaces. In this paper we develop a Bank–Weiser type a posteriori error analysis yielding a reliable estimate and propose the corresponding adaptive algorithm to compute the mixed finite element solutions. Several numerical results illustrating the efficiency of the method are also provided. 2003 Elsevier B.V. All rights reserved. AMS: 65N30; 65N22; 65N15; 76D07; 76M10

On the coupling of boundary integral and mixed finite element methods

We present a numerical method for solving an exterior Dirichlet problem in the plane. The technique consists in coupling boundary integral and mixed finite element methods. An artificial boundary is introduced separating an interior region from an exterior one. From an integral representation of the solution in the exterior domain we deduce two integral equations which relate the solution and its normal derivative over the artificial boundary. These integral equations are incorporated into the so-called mixed formulation of the problem in the interior region and a finite element method is used to approximate the resulting variational problem.

A dual-dual mixed formulation for nonlinear exterior transmission problems

We combine a dual-mixed finite element method with a Dirichletto-Neumann mapping (derived by the boundary integral equation method) to study the solvability and Galerkin approximations of a class of exterior nonlinear transmission problems in the plane. As a model problem, we consider a nonlinear elliptic equation in divergence form coupled with the Laplace equation in an unbounded region of the plane. Our combined approach leads to what we call a dual-dual mixed variational formulation since the main operator involved has itself a dual-type structure. We establish existence and uniqueness of solution for the continuous and discrete formulations, and provide the corresponding error analysis by using Raviart-Thomas elements. The main tool of our analysis is given by a generalization of the usual Babuska-Brezzi theory to a class of nonlinear variational problems with constraints.

Analysis of the Coupling of Primal and Dual-Mixed Finite Element Methods for a Two-Dimensional Fluid-Solid Interaction Problem

This paper deals with a time-harmonic fluid-solid interaction problem posed in the plane. More precisely, we apply the coupling of primal and dual-mixed finite element methods to compute both the pressure of the scattered wave in the linearized fluid and the elastic vibrations that take place in the solid elastic body. To this end, we solve a transmission problem holding between the cross-section of the infinitely long cylinder representing the obstacle and an annular region surrounding it. The novelty of our method lies in the use of a dual-mixed variational formulation in the obstacle, while maintaining the usual primal formulation in the fluid. In other words, we introduce a stress-pressure formulation of the problem instead of the traditional displacement-pressure encountered in the literature. As a consequence, one of the transmission conditions becomes essential, and hence we enforce it weakly by means of a Lagrange multiplier. Next, we apply the abstract framework developed in a recent work by A. Buffa, prove that our coupled variational formulation is well posed, and define the corresponding discrete scheme by using PEERS in the solid domain and standard Lagrange finite elements in the fluid domain. Then we show that the resulting Galerkin scheme is uniquely solvable and convergent and derive optimal error estimates. Finally, we illustrate our analysis with some results from computational experiments.

An Optimal Iterative Process for the Johnson--Nedelec Method of Coupling Boundary and Finite Elements

We reformulate the discretization of the Johnson--Nedelec method [11] of coupling boundary elements and finite elements for an exterior bidimensional Laplacian. This new formulation leads to optimal error estimates and allows the use of simple quadrature formulas for calculation of the boundary element matrix. We show that if the parameter of discretization is sufficiently small, the fully discrete scheme is well posed and the error estimates remain unaltered. The rest of the paper is devoted to the study of an efficient algorithm for solving the resulting discrete linear systems.

A Fully Discrete BEM-FEM for the Exterior Stokes Problem in the Plane

We reformulate the Johnson--Nedelec approach for the exterior two-dimensional Stokes problem taking advantage of the parameterization of the artificial boundary. The main aim of this paper is the presentation and analysis of a fully discrete numerical method for this problem. This one responds to the needs of having efficient approximate quadratures for the weakly singular boundary integrals. We give a complete error analysis of both the Galerkin and fully discrete Galerkin methods.

A residual-based a posteriori error estimator for a two-dimensional fluid–solid interaction problem

In this paper, we develop an a posteriori error analysis of a mixed finite element method for a fluid–solid interaction problem posed in the plane. The media are governed by the acoustic and elastodynamic equations in time-harmonic regime, respectively, and the transmission conditions are given by the equilibrium of forces and the equality of the normal displacements of the solid and the fluid. The coupling of primal and dual-mixed finite element methods is applied to compute both the pressure of the scattered wave in the linearized fluid and the elastic vibrations that take place in the elastic body. The finite element subspaces consider continuous piecewise linear elements for the pressure and a Lagrange multiplier defined on the interface, and PEERS for the stress and rotation in the solid domain. We derive a reliable and efficient residual-based a posteriori error estimator for this coupled problem. Suitable auxiliary problems, the continuous inf-sup conditions satisfied by the bilinear forms involved, a discrete Helmholtz decomposition, and the local approximation properties of the Clément interpolant and Raviart–Thomas operator are the main tools for proving the reliability of the estimator. Then, Helmholtz decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are employed to show the efficiency. Finally, some numerical results confirming the reliability and efficiency of the estimator are reported.

Computing Acoustic Waves in an Inhomogeneous Medium of the Plane by a Coupling of Spectral and Finite Elements

In this paper we analyze a Galerkin procedure, based on a combination of finite and spectral elements, for approximating a time-harmonic acoustic wave scattered by a bounded inhomogeneity. The finite element method used to approximate the near field in the region of inhomogeneity is coupled with a nonlocal boundary condition, which consists in a linear integral equation. This integral equation is discretized by a spectral Galerkin approximation method. We provide error estimates for the Galerkin method, propose fully discrete schemes based on elementary quadrature formulas, and show that the perturbation due to this numerical integration gives rise to a quasi-optimal rate of convergence. We also suggest a method for implementing the algorithm using the preconditioned GMRES method and provide some numerical results.

A mixed–FEM and BEM coupling for a three-dimensional eddy current problem

We study in this paper the electromagnetic field generated in an infinite cylindrical conductor by an alternating current density. The resulting interface problem (see [1]) between the metal and the dielectric medium is treated by a mixed-FEM and BEM coupling method. We prove that our BEM-FEM formulation is well posed and leads to a convergent Galerkin method with optimal error estimates. Furthermore, we introduce a fully discrete version of our Galerkin scheme based on simple quadrature formulas. We show that, if the parameter of discretization is sufficiently small, the fully discrete method is well posed and the error estimates remain unaltered.

An augmented mixed finite element method for 3D linear elasticity problems

In this paper we introduce and analyze a new augmented mixed finite element method for linear elasticity problems in 3D. Our approach is an extension of a technique developed recently for plane elasticity, which is based on the introduction of consistent terms of Galerkin least-squares type. We consider non-homogeneous and homogeneous Dirichlet boundary conditions and prove that the resulting augmented variational formulations lead to strongly coercive bilinear forms. In this way, the associated Galerkin schemes become well posed for arbitrary choices of the corresponding finite element subspaces. In particular, Raviart-Thomas spaces of order 0 for the stress tensor, continuous piecewise linear elements for the displacement, and piecewise constants for the rotation can be utilized. Moreover, we show that in this case the number of unknowns behaves approximately as 9.5 times the number of elements (tetrahedrons) of the triangulation, which is cheaper, by a factor of 3, than the classical PEERS in 3D. Several numerical results illustrating the good performance of the augmented schemes are provided.