Showing papers in "Numerische Mathematik in 2000"

A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem

Abstract: Summary. The $L^2$ Monge-Kantorovich mass transfer problem [31] is reset in a fluid mechanics framework and numerically solved by an augmented Lagrangian method.

Approximation of boundary element matrices

Abstract: This article considers the problem of approximating a general asymptotically smooth function in two variables, typically arising in integral formulations of boundary value problems, by a sum of products of two functions in one variable. From these results an iterative algorithm for the low-rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed. This algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance.

Runge-Kutta methods in optimal control and the transformed adjoint system

Abstract: The convergence rate is determined for Runge-Kutta discretizations of nonlinear control problems. The analysis utilizes a connection between the Kuhn-Tucker multipliers for the discrete problem and the adjoint variables associated with the continuous minimum principle. This connection can also be exploited in numerical solution techniques that require the gradient of the discrete cost function.

Multigrid in H(div) and H(curl)

Abstract: Summary. We consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the Hilbert spaces ${\bf H (div)}$ and ${\bf H (curl)}$ in three dimensions. We show that if appropriate finite element spaces and appropriate additive or multiplicative Schwarz smoothers are used, then the multigrid V-cycle is an efficient solver and preconditioner for the discrete operator. All results are uniform with respect to the mesh size, the number of mesh levels, and weights on the two terms in the inner products.

Residual type a posteriori error estimates for elliptic obstacle problems

Abstract: A posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems. An instrumental ingredient is a new interpolation operator which requires minimal regularity, exhibits optimal approximation properties and preserves positivity. Both upper and lower bounds are proved and their optimality is explored with several examples. Sharp a priori bounds for the a posteriori estimators are given, and extensions of the results to double obstacle problems are briefly discussed.

Adaptive finite element methods for elliptic equations with non-smooth coefficients

Abstract: Nous considerons une equation elliptique du second ordre a coefficients discontinus ou anisotropes dans un domaine borne en dimension 2 ou 3, et sa discretisation par elements finis. Le but de cet article est de demontrer des estimations d'erreur a priori et a posteriori dans une norme appropriee qui soient independantes de la variation des coefficients.

Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval

Abstract: Summary. A Laguerre-Galerkin method is proposed and analyzed for the Burgers equation and Benjamin-Bona-Mahony (BBM) equation on a semi-infinite interval. By reformulating these equations with suitable functional transforms, it is shown that the Laguerre-Galerkin approximations are convergent on a semi-infinite interval with spectral accuracy. An efficient and accurate algorithm based on the Laguerre-Galerkin approximations to the transformed equations is developed and implemented. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented.

On the Convergence of a Dual-Primal Substructuring Method

Abstract: In the Dual-Primal FETI method, introduced by Farhat et al. (1999), 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 $C(1+\log^2(H/h))$ 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.

The condition number of real Vandermonde, Krylov and positive definite Hankel matrices

Abstract: We show that the Euclidean condition number of any positive definite Hankel matrix of order $n\geq 3$ may be bounded from below by $\gamma^{n-1}/(16n)$ with $\gamma=\exp(4 \cdot{\it Catalan}/\pi) \approx 3.210$ , and that this bound may be improved at most by a factor $8 \gamma n$ . Similar estimates are given for the class of real Vandermonde matrices, the class of row-scaled real Vandermonde matrices, and the class of Krylov matrices with Hermitian argument. Improved bounds are derived for the case where the abscissae or eigenvalues are included in a given real interval. Our findings confirm that all such matrices – including for instance the famous Hilbert matrix – are ill-conditioned already for “moderate” order. As application, we describe implications of our results for the numerical condition of various tasks in Numerical Analysis such as polynomial and rational i nterpolation at real nodes, determination of real roots of polynomials, computation of coefficients of orthogonal polynomials, or the iterative solution of linear systems of equations.

A two-level domain decomposition method for the iterative solution of high frequency exterior Helmholtz problems

Abstract: Summary. We present a Lagrange multiplier based two-level domain decomposition method for solving iteratively large-scale systems of equations arising from the finite element discretization of high-frequency exterior Helmholtz problems. The proposed method is essentially an extension of the regularized FETI (Finite Element Tearing and Interconnecting) method to indefinite problems. Its two key ingredients are the regularization of each subdomain matrix by a complex interface lumped mass matrix, and the preconditioning of the interface problem by an auxiliary coarse problem constructed to enforce at each iteration the orthogonality of the residual to a set of carefully chosen planar waves. We show numerically that the proposed method is scalable with respect to the mesh size, the subdomain size, and the wavenumber. We report performance results for a submarine application that highlight the efficiency of the proposed method for the solution of high frequency acoustic scattering problems discretized by finite elements.

A stabilized finite element method for the incompressible magnetohydrodynamic equations

Abstract: We propose and analyze a stabilized finite element method for the incompressible magnetohydrodynamic equations. The numerical results that we present show a good behavior of our approximation in experiments which are relevant from an industrial viewpoint. We explain in particular in the proof of our convergence theorem why it may be interesting to stabilize the magnetic equation as soon as the hydrodynamic diffusion is small and even if the magnetic diffusion is large. This observation is confirmed by our numerical tests.

Implementation of an adaptive finite-element approximation of the Mumford-Shah functional

Abstract: We present and detail a method for the numerical solving of the Mumford-Shah problem, based on a finite element method and on adaptive meshes. We start with the formulation introduced in [13], detail its numerical implementation and then propose a variant which is proved to converge to the Mumford-Shah problem. A few experiments are illustrated.

Nonnegativity preserving convergent schemes for the thin film equation

Abstract: We present numerical schemes for fourth order degenerate parabolic equations that arise e.g. in lubrication theory for time evolution of thin films of viscous fluids. We prove convergence and nonnegativity results in arbitrary space dimensions. A proper choice of the discrete mobility enables us to establish discrete counterparts of the essential integral estimates known from the continuous setting. Hence, the numerical cost in each time step reduces to the solution of a linear system involving a sparse matrix. Furthermore, by introducing a time step control that makes use of an explicit formula for the normal velocity of the free boundary we keep the numerical cost for tracing the free boundary low.

Fortin operator and discrete compactness for edge elements

Abstract: The basic properties of the edge elements are proven in the original papers by Nedelec [22,23] In the two-dimensional case the edge elements are isomorphic to the face elements (the well-known Raviart–Thomas elements [24]), so that all known results concerning face elements can be easily formulated for edge elements In three-dimensional domains this is not the case The aim of the present paper is to show how to construct a Fortin operator which converges uniformly to the identity in the spirit of [5,4] The construction is given for any order tetrahedral edge elements in general geometries We relate this result to the well-known commuting diagram property and apply it to improve the error estimate for a mixed problem which involves edge elements Finally we show that this result can be applied to the analysis of the approximation of the time-harmonic Maxwell's system

Simplicial grid refinement: on Freudenthal's algorithm and the optimal number of congruence classes

Abstract: In the present paper we investigate Freudenthal's simplex refinement algorithm which can be considered to be the canonical generalization of Bank's well known red refinement strategy for triangles. Freudenthal's algorithm subdivides any given (n)-simplex into $2^n$ subsimplices, in such a way that recursive application results in a stable hierarchy of consistent triangulations. Our investigations concentrate in particular on the number of congruence classes generated by recursive refinements. After presentation of the method and the basic ideas behind it, we will show that Freudenthal's algorithm produces at most n!/2 congruence classes for any initial (n)-simplex, no matter how many subsequent refinements are performed. Moreover, we will show that this number is optimal in the sense that recursive application of any affine invariant refinement strategy with $2^n$ sons per element results in at least n!/2 congruence classes for almost all (n)-simplices.

An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes

Abstract: A new a posteriori residual error estimator is defined and rigorously analysed for anisotropic tetrahedral finite element meshes. All considerations carry over to anisotropic triangular meshes with minor changes only. The lower error bound is obtained by means of bubble functions and the corresponding anisotropic inverse inequalities. In order to prove the upper error bound, it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration. To measure this correspondence, a so-called matching function is defined, and its discussion shows it to be a useful tool. With its help anisotropic interpolation estimates and subsequently the upper error bound are proven. Additionally it is pointed out how to treat Robin boundary conditions in a posteriori error analysis on isotropic and anisotropic meshes. A numerical example supports the anisotropic error analysis.

Generalized p-FEM in homogenization

Abstract: Summary. A new finite element method for elliptic problems with locally periodic microstructure of length $\varepsilon >0$ is developed and analyzed. It is shown that the method converges, as $\varepsilon \rightarrow 0$ , to the solution of the homogenized problem with optimal order in $\varepsilon$ and exponentially in the number of degrees of freedom independent of $\varepsilon > 0$ . The computational work of the method is bounded independently of $\varepsilon$ . Numerical experiments demonstrate the feasibility and confirm the theoretical results.

Residual-free bubbles for advection-diffusion problems: the general error analysis

Abstract: We develop the general a priori error analysis of residual-free bubble finite element approximations to linear elliptic convection-dominated diffusion problems subject to homogeneous Dirichlet boundary condition. Optimal-order error bounds are derived in various norms, using piecewise polynomial finite elements of degree greater than or equal to 1.

A minimal stabilisation procedure for mixed finite element methods.

Abstract: Stabilisation methods are often used to circumvent the difficulties associated with the stability of mixed finite element methods. Stabilisation however also means an excessive amount of dissipation or the loss of nice conservation properties. It would thus be desirable to reduce these disadvantages to a minimum. We present a general framework, not restricted to mixed methods, that permits to introduce a minimal stabilising term and hence a minimal perturbation with respect to the original problem. To do so, we rely on the fact that some part of the problem is stable and should not be modified. Sections 2 and 3 present the method in an abstract framework. Section 4 and 5 present two classes of stabilisations for the inf-sup condition in mixed problems. We present many examples, most arising from the discretisation of flow problems. Section 6 presents examples in which the stabilising terms is introduced to cure coercivity problems.

On Landweber iteration for nonlinear ill-posed problems in Hilbert scales

Abstract: In this paper we derive convergence rates results for Landweber iteration in Hilbert scales in terms of the iteration index \(k\) for exact data and in terms of the noise level \(\delta\) for perturbed data. These results improve the one obtained recently for Landweber iteration for nonlinear ill-posed problems in Hilbert spaces. For numerical computations we have to approximate the nonlinear operator and the infinite-dimensional spaces by finite-dimensional ones. We also give a convergence analysis for this finite-dimensional approximation. The conditions needed to obtain the rates are illustrated for a nonlinear Hammerstein integral equation. Numerical results are presented confirming the theoretical ones.

Analysis of a pressure-stabilized finite element approximation of the stationary Navier-Stokes equations

Abstract: The purpose of this paper is to analyze a finite element approximation of the stationary Navier-Stokes equations that allows the use of equal velocity-pressure interpolation. The idea is to introduce as unknown of the discrete problem the projection of the pressure gradient (multiplied by suitable algorithmic parameters) onto the space of continuous vector fields. The difference between these two vectors (pressure gradient and projection) is introduced in the continuity equation. The resulting formulation is shown to be stable and optimally convergent, both in a norm associated to the problem and in the \(L^2\) norm for both velocities and pressure. This is proved first for the Stokes problem, and then it is extended to the nonlinear case. All the analysis relies on an inf-sup condition that is much weaker than for the standard Galerkin approximation, in spite of the fact that the present method is only a minor modification of this.

Wavelet adaptive method for second order elliptic problems: boundary conditions and domain decomposition

Abstract: Summary. Wavelet methods allow to combine high order accuracy, multilevel preconditioning techniques and adaptive approximation, in order to solve efficiently elliptic operator equations. One of the main difficulty in this context is the efficient treatment of non-homogeneous boundary conditions. In this paper, we propose a strategy that allows to append such conditions in the setting of space refinement (i.e. adaptive) discretizations of second order problems. Our method is based on the use of compatible multiscale decompositions for both the domain and its boundary, and on the possibility of characterizing various function spaces from the numerical properties of these decompositions. In particular, this allows the construction of a lifting operator which is stable for a certain range of smoothness classes, and preserves the compression of the solution in the wavelet basis. An explicit construction of the wavelet bases and the lifting is proposed on fairly general domains, based on $C^0$ conforming domain decomposition techniques.

A finite element solution of an added mass formulation for coupled fluid-solid vibrations

Abstract: A finite element method to approximate the vibration modes of a structure in contact with an incompressible fluid is analyzed in this paper. The effect of the fluid is taken into account by means of an added mass formulation, which is one of the most usual procedures in engineering practice. Gravity waves on the free surface of the liquid are also considered in the model. Piecewise linear continuous elements are used to discretize the solid displacements, the variables to compute the added mass terms and the vertical displacement of the free surface, yielding a non conforming method for the spectral coupled problem. Error estimates are settled for approximate eigenfunctions and eigenfrequencies. Implementation issues are discussed and numerical experiments are reported. In particular the method is compared with other numerical scheme, based on a pure displacement formulation, which has been recently analyzed.

Nonlinear functionals of wavelet expansions – adaptive reconstruction and fast evaluation

Abstract: Summary. This paper is concerned with the efficient evaluation of nonlinear expressions of wavelet expansions obtained through an adaptive process. In particular, evaluation covers here the computation of inner products of such expressions with wavelets which arise, for instance, in the context of Galerkin or Petrov Galerkin schemes for the solution of differential equations. The central objective is to develop schemes that facilitate such evaluations at a computational expense exceeding the complexity of the given expansion, i.e., the number of nonzero wavelet coefficients, as little as possible. The following issues are addressed. First, motivated by previous treatments of the subject, we discuss the type of regularity assumptions that are appropriate in this context and explain the relevance of Besov norms. The principal strategy is to relate the computation of inner products of wavelets with compositions to approximations of compositions in terms of possibly few dual wavelets. The analysis of these approximations finally leads to a concrete evaluation scheme which is shown to be in a certain sense asymptotically optimal. We conclude with a simple numerical example.

Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes

Abstract: Summary. Both for the $H^1$ - and $L^2$ -norms, we prove that, up to higher order perturbation terms, edge residuals yield global upper and local lower bounds on the error of linear finite element methods on anisotropic triangular or tetrahedral meshes. We also show that, with a correct scaling, edge residuals yield a robust error estimator for a singularly perturbed reaction-diffusion equation.

Numerical solution of constant coefficient linear delay differential equations as abstract Cauchy problems

Abstract: Summary. In this paper we present an approach for the numerical solution of delay differential equations \begin{equation} \left\{ \begin{array}{l} y^{\prime }\left( t\right) =Ly\left( t\right) +My\left( t-\tau \right) \;\;t\geq 0 y\left( t\right) =\varphi \left( t\right) \;\;-\tau \leq t\leq 0, \end{array} \right. \end{equation} where $\tau >0$ , $L,M\in \mathbb{C}^{m\times m}$ and $\varphi \in C\left( \left[ -\tau ,0\right] ,\mathbb{C}^m\right) $ , different from the classical step-by-step method. We restate (1) as an abstract Cauchy problem and then we discretize it in a system of ordinary differential equations. The scheme of discretization is proved to be convergent. Moreover the asymptotic stability is investigated for two significant classes of asymptotically stable problems (1).

Optimal estimates for lower and upper bounds of approximation errors in the p-version of the finite element method in two dimensions.

Abstract: This paper will analyze the lower and upper error bounds of the finite element solution of the p-version for linear elliptic problems in polygonal domains. The optimal rate of convergence is rigorously proved based on the sharp estimates of lower and upper bounds of the approximation error.

Numerical approximation of Young measures in non-convex variational problems

Abstract: Summary. In non-convex optimisation problems, in particular in non-convex variational problems, there usually does not exist any classical solution but only generalised solutions which involve Young measures. In this paper, first a suitable relaxation and approximation theory is developed together with optimality conditions, and then an adaptive scheme is proposed for the efficient numerical treatment. The Young measures solving the approximate problems are usually composed only from a few atoms. This is the main argument our effective active-set type algorithm is based on. The support of those atoms is estimated from the Weierstrass maximum principle which involves a Hamiltonian whose good guess is obtained by a multilevel technique. Numerical experiments are performed in a one-dimensional variational problem and support efficiency of the algorithm.

Numerical analysis of a linearised fluid-structure interaction problem

Abstract: 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.

Effective preconditioning of Uzawa type schemes for a generalized Stokes problem

Abstract: The Schur complement of a model problem is considered as a preconditioner for the Uzawa type schemes for the generalized Stokes problem (the Stokes problem with the additional term \(\alpha\boldmath{u}\) in the motion equation). The implementation of the preconditioned method requires for each iteration only one extra solution of the Poisson equation with Neumann boundary conditions. For a wide class of 2D and 3D domains a theorem on its convergence is proved. In particular, it is established that the method converges with a rate that is bounded by some constant independent of \(\alpha\). Some finite difference and finite element methods are discussed. Numerical results for finite difference MAC scheme are provided.