The Mortar finite element method with Lagrange multipliers

Summary. The present paper deals with a variant of a non conforming domain decomposition technique: the mortar finite element method. In the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices (and the edges in 3D) of subdomains. It is shown that, written under primal hybrid formulation, the approximation problem, issued from a discretization of a second order elliptic equation in 2D, is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms. Finally the parallelization advantages consequence of this variant are also addressed. Received December 1, 1996 / Revised version received November 23, 1998 / Published online September 24, 1999     

Appending boundary conditions by Lagrange multipliers: Analysis of the LBB condition

Summary. This paper is concerned with the analysis of discretization schemes for second order elliptic boundary value problems when essential boundary conditions are enforced with the aid of Lagrange multipliers. Specifically, we show how the validity of the Ladyškaja–Babušska–Brezzi (LBB) condition for the corresponding saddle point problems depends on the various ingredients of the involved discretizations. The main result states that the LBB condition is satisfied whenever the discretization step length on the boundary, , is somewhat bigger than the one on the domain, . This is quantified through constants stemming from the trace theorem, norm equivalences for the multiplier spaces on the boundary, and direct and inverse inequalities. In order to better understand the interplay of these constants, we then specialize the setting to wavelet discretizations. In this case the stability criteria can be stated solely in terms of spectral properties of wavelet representations of the trace operator. We conclude by illustrating our theoretical findings by some numerical experiments. We stress that the results presented here apply to any spatial dimension and to a wide selection of Lagrange multiplier spaces which, in particular, need not be traces of the trial spaces. However, we do always assume that a hierarchy of nested trial spaces is given. Received October 23, 1998 / Revised version received December 27, 1999 / Published online October 16, 2000     

Wavelet stabilization of the Lagrange multiplier method

Summary. We propose here a stabilization strategy for the Lagrange multiplier formulation of Dirichlet problems. The stabilization is based on the use of equivalent scalar products for the spaces and , which are realized by means of wavelet functions. The resulting stabilized bilinear form is coercive with respect to the natural norm associated to the problem. A uniformly coercive approximation of the stabilized bilinear form is constructed for a wide class of approximation spaces, for which an optimal error estimate is provided. Finally, a formulation is presented which is obtained by eliminating the multiplier by static condensation. This formulation is closely related to the Nitsche's method for solving Dirichlet boundary value problems. Received December 4, 1998 / Revised version received May 7, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

Convergence of a finite element method for non-parametric mean curvature flow

Summary. Convergence for the spatial discretization by linear finite elements of the non-parametric mean curvature flow is proved under natural regularity assumptions on the continuous solution. Asymptotic convergence is also obtained for the time derivative which is proportional to mean curvature. An existence result for the continuous problem in adequate spaces is included. Received September 30, 1993     

On the convergence of a dual-primal substructuring method

Summary.   In the Dual-Primal FETI method, introduced by Farhat et al. [5], the domain is decomposed into non-overlapping subdomains, but the degrees of freedom on crosspoints remain common to all subdomains adjacent to the crosspoint. The continuity of the remaining degrees of freedom on subdomain interfaces is enforced by Lagrange multipliers and all degrees of freedom are eliminated. The resulting dual problem is solved by preconditioned conjugate gradients. We give an algebraic bound on the condition number, assuming only a single inequality in discrete norms, and use the algebraic bound to show that the condition number is bounded by for both second and fourth order elliptic selfadjoint problems discretized by conforming finite elements, as well as for a wide class of finite elements for the Reissner-Mindlin plate model. Received January 20, 2000 / Revised version received April 25, 2000 / Published online December 19, 2000     

Stability and analyticity in maximum-norm for simplicial Lagrange finite element semidiscretizations of parabolic equations with Dirichlet boundary conditions

Stability and analyticity estimates in maximum-norm are shown for spatially discrete finite element approximations based on simplicial Lagrange elements for the model heat equation with Dirichlet boundary conditions. The bounds are logarithm free and valid in arbitrary dimension and for arbitrary polynomial degree. The work continues an earlier study by Schatz et al. [5] in which Neumann boundary conditions were considered. Received November 1998 / Revised version received August 11, 1999 / Published online July 12, 2000     

Convergence of the infinite element methods for the Helmholtz equation in separable domains

To the best knowledge of the authors, this work presents the first convergence analysis for the Infinite Element Method (IEM) for the Helmholtz equation in exterior domains. The approximation applies to separable geometries only, combining an arbitrary Finite Element (FE) discretization on the boundary of the domain with a spectral-like approximation in the “radial” direction, with shape functions resulting from the separation of variables. The principal idea of the presented analysis is based on the spectral decomposition of the problem. Received February 10, 1996 / Revised version received February 17, 1997     

Analysis of a coupled finite-infinite element method for exterior Helmholtz problems

Summary. This analysis of convergence of a coupled FEM-IEM is based on our previous work on the FEM and the IEM for exterior Helmholtz problems. The key idea is to represent both the exact and the numerical solution by the Dirichlet-to-Neumann operators that they induce on the coupling hypersurface in the exterior of an obstacle. The investigation of convergence can then be related to a spectral analysis of these DtN operators. We give a general outline of our method and then proceed to a detailed investigation of the case that the coupling surface is a sphere. Our main goal is to explore the convergence mechanism. In this context, we show well-posedness of both the continuous and the discrete models. We further show that the discrete inf-sup constants have a positive lower bound that does not depend on the number of DOF of the IEM. The proofs are based on lemmas on the spectra of the continuous and the discrete DtN operators, where the spectral characterization of the discrete DtN operator is given as a conjecture from numerical experiments. In our convergence analysis, we show algebraic (in terms of N) convergence of arbitrary order and generalize this result to exponential convergence. Received April 10, 1999 / Revised version received November 10, 1999 / Published online October 16, 2000     

A sparse finite element method with high accuracy Part I

Summary. In this paper, we develop and analyze a new finite element method called the sparse finite element method for second order elliptic problems. This method involves much fewer degrees of freedom than the standard finite element method. We show nevertheless that such a sparse finite element method still possesses the superconvergence and other high accuracy properties same as those of the standard finite element method. The main technique in our analysis is the use of some integral identities. Received October 1, 1995 / Revised version received August 23, 1999 / Published online February 5, 2001     

An unusual stabilized finite element method for a generalized Stokes problem

Summary. An unusual stabilized finite element is presented and analyzed herein for a generalized Stokes problem with a dominating zeroth order term. The method consists in subtracting a mesh dependent term from the formulation without compromising consistency. The design of this mesh dependent term, as well as the stabilization parameter involved, are suggested by bubble condensation. Stability is proven for any combination of velocity and pressure spaces, under the hypotheses of continuity for the pressure space. Optimal order error estimates are derived for the velocity and the pressure, using the standard norms for these unknowns. Numerical experiments confirming these theoretical results, and comparisons with previous methods are presented. Received April 26, 2001 / Revised version received July 30, 2001 / Published online October 17, 2001 Correspondence to: Gabriel R. Barrenechea     

The maximum angle condition in the finite element method for monotone problems with applications in magnetostatics

Summary. The finite element method for an elliptic equation with discontinuous coefficients (obtained for the magnetic potential from Maxwell's equations) is analyzed in the union of closed domains the boundaries of which form a system of three circles with the same centre. As the middle domain is very narrow the triangulations obeying the maximum angle condition are considered. In the case of piecewise linear trial functions the maximum rate of convergence in the norm of the space is proved under the following conditions: 1. the exact solution is piecewise of class ; 2. the family of subtriangulations of the narrow subdomain satisfies the maximum angle condition expressed by relation (38). The paper extends the results of [24]. Received March 8, 1993 / Revised version received November 28, 1994     

An iterative substructuring method for the p-version of the boundary element method for hypersingular integral operators in three dimensions

Summary. We study preconditioners for the -version of the boundary element method for hypersingular integral equations in three dimensions. The preconditioners are based on iterative substructuring of the underlying ansatz spaces which are constructed by using discretely harmonic basis functions. We consider a so-called wire basket preconditioner and a non-overlapping additive Schwarz method based on the complete natural splitting, i.e. with respect to the nodal, edge and interior functions, as well as an almost diagonal preconditioner. In any case we add the space of piecewise bilinear functions which eliminate the dependence of the condition numbers on the mesh size. For all these methods we prove that the resulting condition numbers are bounded by . Here, is the polynomial degree of the ansatz functions and is a constant which is independent of and the mesh size of the underlying boundary element mesh. Numerical experiments supporting these results are reported. Received July 8, 1996 / Revised version received January 8, 1997     

Finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy

Summary. A model for the phase separation of a multi-component alloy with non-smooth free energy is considered. An error bound is proved for a fully practical piecewise linear finite element approximation using a backward Euler time discretization. An iterative scheme for solving the resulting nonlinear algebraic system is analysed. Finally numerical experiments with three components in one and two space dimensions are presented. In the one dimensional case we compare some steady states obtained numerically with the corresponding stationary solutions of the continuous problem, which we construct explicitly. Received September 28, 1995 / Revised version received May 6, 1996     

Substructuring preconditioners for the Q_1 mortar element method

Summary. The mortar element method is a non conforming finite element method with elements based on domain decomposition. For the Laplace equation, it yields an ill conditioned linear system. For solving the linear system, the so called preconditioned conjugate gradient method in a subspace is used. Preconditioners are proposed, and estimates on condition numbers and arithmetical complexity are given. Finally, numerical experiments are presented. Received June 22, 1994 / Revised version received February 6, 1995     

The full approximation accuracy for the stream function-vorticity-pressure method

Summary. By means of identity techniques, in this paper, we develop the stream function-vorticity-pressure method and obtain the full approximation convergence and global superconvergence estimates for the Stokes equations. Received September 3, 1992 / Revised version received February 1, 1994     

Bounds on spectral condition numbers of matrices arising in the p-version of the finite element method

Summary. We estimate condition numbers of -version matrices for tensor product elements with two choices of reference element degrees of freedom. In one case (Lagrange elements) the condition numbers grow exponentially in , whereas in the other (hierarchical basis functions based on Tchebycheff polynomials) the condition numbers grow rapidly but only algebraically in . We conjecture that regardless of the choice of basis the condition numbers grow like or faster, where is the dimension of the spatial domain. Received August 8, 1992 / Revised version received March 25, 1994     

Multigrid algorithms for a vertex–centered covolume method for elliptic problems

Summary. We analyze V–cycle multigrid algorithms for a class of perturbed problems whose perturbation in the bilinear form preserves the convergence properties of the multigrid algorithm of the original problem. As an application, we study the convergence of multigrid algorithms for a covolume method or a vertex–centered finite volume element method for variable coefficient elliptic problems on polygonal domains. As in standard finite element methods, the V–cycle algorithm with one pre-smoothing converges with a rate independent of the number of levels. Various types of smoothers including point or line Jacobi, and Gauss-Seidel relaxation are considered. Received August 19, 1999 / Revised version received July 10, 2000 / Published online June 7, 2001     

On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow

Summary. The aim of this work is to study a decoupled algorithm of a fixed point for solving a finite element (FE) problem for the approximation of viscoelastic fluid flow obeying an Oldroyd B differential model. The interest for this algorithm lies in its applications to numerical simulation and in the cost of computing. Furthermore it is easy to bring this algorithm into play. The unknowns are the viscoelastic part of the extra stress tensor, the velocity and the pressure. We suppose that the solution is sufficiently smooth and small. The approximation of stress, velocity and pressure are resp. discontinuous, continuous, continuous FE. Upwinding needed for convection of , is made by discontinuous FE. The method consists to solve alternatively a transport equation for the stress, and a Stokes like problem for velocity and pressure. Previously, results of existence of the solution for the approximate problem and error bounds have been obtained using fixed point techniques with coupled algorithm. In this paper we show that the mapping of the decoupled fixed point algorithm is locally (in a neighbourhood of ) contracting and we obtain existence, unicity (locally) of the solution of the approximate problem and error bounds. Received July 29, 1994 / Revised version received March 13, 1995     

Numerical analysis of a linearised fluid-structure interaction problem

Summary. This paper describes the numerical analysis of a time dependent linearised fluid structure interaction problems involving a very viscous fluid and an elastic shell in small displacements. For simplicity, all changes of geometry are neglected. A single variational formulation is proposed for the whole problem and generic discretisation strategies are introduced independently on the fluid and on the structure. More precisely, the space approximation of the fluid problem is realized by standard mixed finite elements, the shell is approximated by DKT finite elements, and time derivatives are approximated either by midpoint rules or by backward difference formula. Using fundamental energy estimates on the continuous problem written in a proper functional space, on its discrete equivalent, and on an associated error evolution equation, we can prove that the proposed variational problem is well posed, and that its approximation in space and time converges with optimal order to the continuous solution. Received May 14, 1999 / Revised version revised October 14, 1999 / Published online July 12, 2000     

Mixed finite element approximation of a degenerate elliptic problem

Summary. We present a mixed finite element approximation of an elliptic problem with degenerate coefficients, arising in the study of the electromagnetic field in a resonant structure with cylindrical symmetry. Optimal error bounds are derived. Received May 4, 1994 / Revised version received September 27, 1994