Showing papers in "Numerische Mathematik in 1998"

Finite element methods and their convergence for elliptic and parabolic interface problems

Abstract: In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems in two-dimensional convex polygonal domains. Nearly the same optimal $L^2$ -norm and energy-norm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth, though the regularities of the solutions are low on the whole domain. The assumptions on the finite element triangulation are reasonable and practical.

Convergence of Algebraic Multigrid Based on Smoothed Aggregation

Abstract: We prove aconvergence estimate for the Algebraic Multigrid Method with prolongations defined by aggregation using zero energy modes, followed by a smoothing. The method input is the problem matrix and a matrix of the zero energy modes. The estimate depends only polylogarithmically on the mesh size, and requires only a weak approximation property for the aggregates, which can be a-priori verified computationally. Construction of the prolongator in the case of a general second order system is described, and the assumptions of the theorem are verified for a scalar problem discretized by linear conforming finite elements.

A posteriori error estimators for convection-diffusion equations

Abstract: We derive a posteriori error estimators for convection-diffusion equations with dominant convection. The estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local mesh-Peclet number. The estimators are either based on the evaluation of local residuals or on the solution of discrete local Dirichlet or Neumann problems.

Third order nonoscillatory central scheme for hyperbolic conservation laws

Abstract: A third-order accurate Godunov-type scheme for the approximate solution of hyperbolic systems of conservation laws is presented. Its two main ingredients include: 1. A non-oscillatory piecewise-quadratic reconstruction of pointvalues from their given cell averages; and 2. A central differencing based on staggered evolution of the reconstructed cell averages. This results in a third-order central scheme, an extension along the lines of the second-order central scheme of Nessyahu and Tadmor \cite{NT}. The scalar scheme is non-oscillatory (and hence – convergent), in the sense that it does not increase the number of initial extrema (– as does the exact entropy solution operator). Extension to systems is carried out by componentwise application of the scalar framework. In particular, we have the advantage that, unlike upwind schemes, no (approximate) Riemann solvers, field-by-field characteristic decompositions, etc., are required. Numerical experiments confirm the high-resolution content of the proposed scheme. Thus, a considerable amount of simplicity and robustness is gained while retaining the expected third-order resolution.

On the approximation of the unsteady Navier–Stokes equations by finite element projection methods

Abstract: This paper provides an analysis of a fractional-step projection method to compute incompressible viscous flows by means of finite element approximations. The analysis is based on the idea that the appropriate functional setting for projection methods must accommodate two different spaces for representing the velocity fields calculated respectively in the viscous and the incompressible half steps of the method. Such a theoretical distinction leads to a finite element projection method with a Poisson equation for the incremental pressure unknown and to a very practical implementation of the method with only the intermediate velocity appearing in the numerical algorithm. Error estimates in finite time are given. An extension of the method to a problem with unconventional boundary conditions is also considered to illustrate the flexibility of the proposed method.

A numerically stable, structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils

Abstract: A new method is presented for the numerical computation of the generalized eigenvalues of real Hamiltonian or symplectic pencils and matrices. The method is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic). In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order \(\sqrt{\varepsilon}\), where \(\varepsilon \) is the machine precision, the new method computes the eigenvalues to full possible accuracy.

Robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation

Abstract: We derive robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation. Here, robust means that the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds is bounded from below and from above by constants which do neither depend on any meshsize nor on the perturbation parameter. The estimators are based either on the evaluation of local residuals or on the solution of discrete local Dirichlet or Neumann problems.

A posteriori error estimates for mixed FEM in elasticity

Abstract: A residue based reliable and efficient error estimator is established for finite element solutions of mixed boundary value problems in linear, planar elasticity. The proof of the reliability of the estimator is based on Helmholtz type decompositions of the error in the stress variable and a duality argument for the error in the displacements. The efficiency follows from inverse estimates. The constants in both estimates are independent of the Lame constant $\lambda$ , and so locking phenomena for $\lambda\to\infty$ are properly indicated. The analysis justifies a new adaptive algorithm for automatic mesh–refinement.

A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinearill-posed problems

Abstract: This paper treats a class of Newton type methods for the approximate solution of nonlinear ill-posed operator equations, that use so-called filter functions for regularizing the linearized equation in each Newton step. For noisy data we derive an aposteriori stopping rule that yields convergence of the iterates to asolution, as the noise level goes to zero, under certain smoothness conditions on the nonlinear operator. Appropriate closeness and smoothness assumptions on the starting value and the solution are shown to lead to optimal convergence rates. Moreover we present an application of the Newton type methods under consideration to a parameter identification problem, together with some numerical results.

A mathematical investigation of the Car-Parrinello method

Abstract: Summary. The Car-Parrinello method for ab-initio molecular dynamics avoids the explicit minimization of energy functionals given by functional density theory in the context of the quantum adiabatic approximation (time-dependent BornOppenheimer approximation). Instead, it introduces a fictitious classical dynamics for the electronic orbitals. For many realistic systems this concept allowed firstprinciple computer simulations for the first time. In this paper we study the quantitative influence of the involved parameter , the fictitious electronic mass of the method. In particular, we prove by use of a carefully chosen two-time-scale asymptotics that the deviation of the Car-Parrinello method from the adiabatic model is of order O ( 1=2 ) ‐ provided one starts in the ground state of the electronic system and the electronic excitation spectrum satisfies a certain nondegeneracy condition. Analyzing a two-level model problem we prove that our result cannot be improved in general.

Regular solutions of nonlinear differential-algebraic equations and their numerical determination

Abstract: For a general class of nonlinear (possibly higher index) differential-algebraic equations we show existence and uniqueness of solutions. These solutions are regular in the sense that Newton's method will converge locally and quadratically. On the basis of the presented theoretical results, numerical methods for the determination of consistent initial values and for the computation of regular solutions are developed. Several numerical examples are included.

Convergence of multidimensional cascade algorithm

Abstract: A necessary and sufficient condition on the spectrum of the restricted transition operator is given for the convergence in $L^2({\Bbb R}^d)$ of the multidimensional cascade algorithm for any starting function $\phi_0$ whose shifts form a partition of unity.

Numerical Taylor expansions of invariant manifolds in large dynamical systems

Abstract: In this paper we develop a numerical method for computing higher order local approximations of invariant manifolds, such as stable, unstable or center manifolds near steady states of a dynamical system. The underlying system is assumed to be large in the sense that a large sparse Jacobian at the equilibrium occurs, for which only a linear (black box) solver and a low dimensional invariant subspace is available, but for which methods like the QR–Algorithm are considered to be too expensive. Our method is based on an analysis of the multilinear Sylvester equations for the higher derivatives which can be solved under certain nonresonance conditions. These conditions are weaker than the standard gap conditions on the spectrum which guarantee the existence of the invariant manifold. The final algorithm requires the solution of several large linear systems with a bordered Jacobian. To these systems we apply a block elimination method recently developed by Govaerts and Pryce [12, 14].

Convergence and stability of finite element nonlinear Galerkin method for the Navier-Stokes equations

Abstract: In this article, we present a new fully discrete finite element nonlinear Galerkin method, which are well suited to the long time integration of the Navier-Stokes equations. Spatial discretization is based on two-grid finite element technique; time discretization is based on Euler explicit scheme with variable time step size. Moreover, we analyse the boundedness, convergence and stability condition of the finite element nonlinear Galerkin method. Our discussion shows that the time step constraints of the method depend only on the coarse grid parameter $H$ and the time step constraints of the finite element Galerkin method depend on the fine grid parameter $h « H$ under the same convergence accuracy.

Fast parallel solvers for symmetric boundary element domain decomposition equations

Abstract: The boundary element method (BEM) is of advantage in many applications including far-field computations in magnetostatics and solid mechanics as well as accurate computations of singularities. Since the numerical approximation is essentially reduced to the boundary of the domain under consideration, the mesh generation and handling is simpler than, for example, in a finite element discretization of the domain. In this paper, we discuss fast solution techniques for the linear systems of equations obtained by the BEM (BE-equations) utilizing the non-overlapping domain decomposition (DD). We study parallel algorithms for solving large scale Galerkin BE–equations approximating linear potential problems in plane, bounded domains with piecewise homogeneous material properties. We give an elementary spectral equivalence analysis of the BEM Schur complement that provides the tool for constructing and analysing appropriate preconditioners. Finally, we present numerical results obtained on a massively parallel machine using up to 128 processors, and we sketch further applications to elasticity problems and to the coupling of the finite element method (FEM) with the boundary element method. As shown theoretically and confirmed by the numerical experiments, the methods are of $O(h^{-2})$ algebraic complexity and of high parallel efficiency, where $h$ denotes the usual discretization parameter.

A term by term stabilization algorithm for finite element solution of incompressible flow problems

Abstract: This paper introduces a stabilization technique for Finite Element numerical solution of 2D and 3D incompressible flow problems. It may be applied to stabilize the discretization of the pressure gradient, and also of any individual operator term such as the convection, curl or divergence operators, with specific levels of numerical diffusion for each one of them. Its computational complexity is reduced with respect to usual (residual-based) stabilization techniques. We consider piecewise affine Finite Elements, for which we obtain optimal error bounds for steady Navier-Stokes and also for generalized Stokes equations (including convection). We include some numerical experiment in well known 2D test cases, that show its good performances.

Determination of reflector surfaces from near-field scattering data II. Numerical solution

Abstract: Let $O$ be a nonisotropic point source of light, and $I(m)$ the power intensity of this source in direction $m$ . Suppose that the light rays emitted by the source $O$ through an aperture $D$ fall on a perfectly reflecting surface $R$ and reflect off it so that the reflected rays illuminate a closed domain $T$ on some plane with intensity $L$ . The inverse problem consists of constructing the reflector surface $R$ from given position of the source $O$ , the input aperture $D$ , function $I$ , “target” set $T$ , and output intensity $L$ . For example, the input intensity may have a “bell”-like shape and we may wish to redistribute the energy uniformly over a prespecified region. The analytical formulation of the described above problem leads to a non-linear partial differential equation of Monge-Ampere type. In our previous paper we proved existence of weak solutions to this inverse problem and in this paper we describe and illustrate with examples an algorithm for its numerical solution. The proposed numerical method can be easily modified for the case when $T$ is a closed domain on an arbitrary surface.

Multipole expansions for the Fokker-Planck-Landau operator

Abstract: We give a sequence of operators approximating the Fokker-Planck-Landau collision operator. This sequence is obtained by aplying the fast multipole method based on the work by Greengard and Rokhlin [17], and tends to the exact Fokker-Planck-Landau operator with an arbitrary accuracy. These operators satisfy the physical properties such as the conservation of mass, momentum, energy and the decay of the entropy. Furthermore, the quadratic structure due to the velocity coupling in the expression of the Fokker-Planck-Landau operator is removed in the approximating operators. This fact reduces seriously the computationnal cost of numerical simulations of the Fokker-Planck-Landau equation. Finally, we give numerical conservative and entropy discretizations solving the homogeneous Fokker-Planck-Landau equation using the fast multipole method. In addition to the deterministic character of these approximations, they give satisfactory results in terms of accuracy and CPU time.

Secant methods for semismooth equations

Abstract: Some generalizations of the secant method to semismooth equations are presented. In the one-dimensional case the superlinear convergence of the classical secant method for general semismooth equations is proved. Moreover a new quadratically convergent method is proposed that requires two function values per iteration. For the n-dimensional cases, we discuss secant methods for two classes of composite semismooth equations. Most often studied semismooth equations are of such form.

A dual graph-norm refinement indicator for finite volume approximations of the Euler equations

Abstract: We generalise and apply a refinement indicator of the type originally designed by Mackenzie, Suli and Warnecke in [15] and [16] for linear Friedrichs systems to the Euler equations of inviscid, compressible fluid flow. The Euler equations are symmetrized by means of entropy variables and locally linearized about a constant state to obtain a symmetric hyperbolic system to which an a posteriori error analysis of the type introduced in [15] can be applied. We discuss the details of the implementation of the refinement indicator into the DLR-\(\tau\)-Code which is based on a finite volume method of box type on an unstructured grid and present numerical results.

Extrapolation techniques for ill-conditioned linear systems

Abstract: In this paper, the regularized solutions of an ill–conditioned system of linear equations are computed for several values of the regularization parameter $\lambda$ . Then, these solutions are extrapolated at $\lambda=0$ by various vector rational extrapolations techniques built for that purpose. These techniques are justified by an analysis of the regularized solutions based on the singular value decomposition and the generalized singular value decomposition. Numerical results illustrate the effectiveness of the procedures.

Stable three-point wavelet bases on general meshes

Abstract: Recently, we introduced a wavelet basis on general, possibly locally refined linear finite element spaces. Each wavelet is a linear combination of three nodal basis functions, independently of the number of space dimensions. In the present paper, we show \(H^s\)-stability of this basis for a range of \(s\), that in any case includes \(s=1\), which means that the corresponding additive Schwarz preconditioner is optimal for second order problems. Furthermore, we generalize the construction of the wavelet basis to manifolds. We show that the wavelets have at least one-, and in areas where the manifold is smooth and the mesh is uniform even two vanishing moments. Because of these vanishing moments, apart from preconditioning, the basis can be used for compression purposes: For a class of integral equation problems, the stiffness matrix with respect to the wavelet basis will be close to a sparse one, in the sense that, a priori, it can be compressed to a sparse matrix without the order of convergence being reduced.

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

Abstract: 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.

Using approximate inverses in algebraic multilevel methods

Abstract: This paper deals with the iterative solution of large sparse symmetric positive definite systems. We investigate preconditioning techniques of the two-level type that are based on a block factorization of the system matrix. Whereas the basic scheme assumes an exact inversion of the submatrix related to the first block of unknowns, we analyze the effect of using an approximate inverse instead. We derive condition number estimates that are valid for any type of approximation of the Schur complement and that do not assume the use of the hierarchical basis. They show that the two-level methods are stable when using approximate inverses based on modified ILU techniques, or explicit inverses that meet some row-sum criterion. On the other hand, we bring to the light that the use of standard approximate inverses based on convergent splittings can have a dramatic effect on the convergence rate. These conclusions are numerically illustrated on some examples

An iterative multi level algorithm for solving nonlinear ill–posed problems

Abstract: The convergence analysis of Landweber's iteration for the solution of nonlinear ill–posed problem has been developed recently by Hanke, Neubauer and Scherzer. In concrete applications, sufficient conditions for convergence of the Landweber iterates developed there (although quite natural) turned out to be complicated to verify analytically. However, in numerical realizations, when discretizations are considered, sufficient conditions for local convergence can usually easily be stated. This paper is motivated by these observations: Initially a discretization is fixed and a discrete Landweber iteration is implemented until an appropriate stopping criterion becomes active. The output is used as an initial guess for a finer discretization. An advantage of this method is that the convergence analysis can be considered in a family of finite dimensional spaces. The numerical performance of this multi level algorithm is compared with Landweber's iteration.

The Euler-Maclaurin expansion and finite-part integrals

Abstract: In this paper we compare G(p), the Mellin transform (together with its analytic continuation), and $\overline{\overline{G}}(p)$ , the related Hadamard finite-part integral of a function g(x), which decays exponentially at infinity and has specified singular behavior at the origin. Except when p is a nonpositive integer, these coincide. When p is a nonpositive integer, $\overline{\overline{G}}(p)$ is well defined, but G(p) has a pole. We show that the terms in the Laurent expansion about this pole can be simply expressed in terms of the Hadamard finite-part integral of a related function. This circumstance is exploited to provide a conceptually uniform proof of the various generalizations of the Euler-Maclaurin expansion for the quadrature error functional.

Local a-posteriori error indicators for the Galerkin discretization of boundary integral equations

Abstract: In this paper we present local a-posteriori error indicators for the Galerkin discretization of boundary integral equations. These error indicators are introduced and investigated by Babuska-Rheinboldt [3] for finite element methods. We transfer them from finite element methods onto boundary element methods and show that they are reliable and efficient for a wide class of integral operators under relatively weak assumptions. These local error indicators are based on the computable residual and can be used for controlling the adaptive mesh refinement.

Estimateur d'erreur a posteriori hiérarchique. Application aux éléments finis mixtes

Abstract: On propose une generalisation de l'estimateur a posteriori hierarchique de Bank-Weiser aux formulation mixtes, dans le cas d'approximations conformes et non conformes, avec ou sans integration numerique. On presente comme exemples d'application: la formulation mixte duale du probleme de Dirichlet pour le Laplacien, avec et sans integration numerique.

Asymptotic expansions of the error of spline Galerkin boundary element methods

Abstract: In this paper we analyse the existence of asymptotic expansions of the error of Galerkin methods with splines of arbitrary degree for the approximate solution of integral equations with logarithmic kernels. These expansions are obtained in terms of an interpolation operator and are useful for the application of Richardson extrapolation and for obtaining sharper error bounds. We also present and analyse a family of fully discrete spline Galerkin methods for the solution of the same equations. Following the analysis of Galerkin methods, we show the existence of asymptotic expansions of the error.

Generalized block Lanczos methods for large unsymmetric eigenproblems

Abstract: Generalized block Lanczos methods for large unsymmetric eigenproblems are presented, which contain the block Arnoldi method, and the block Arnoldi algorithms are developed The convergence of this class of methods is analyzed when the matrix A is diagonalizable Upper bounds for the distances between normalized eigenvectors and a block Krylov subspace are derived, and a priori theoretical error bounds for Ritz elements are established Compared with generalized Lanczos methods, which contain Arnoldi's method, the convergence analysis shows that the block versions have two advantages: First, they may be efficient for computing clustered eigenvalues; second, they are able to solve multiple eigenproblems However, a deep analysis exposes that the approximate eigenvectors or Ritz vectors obtained by general orthogonal projection methods including generalized block methods may fail to converge theoretically for a general unsymmetric matrix A even if corresponding approximate eigenvalues or Ritz values do, since the convergence of Ritz vectors needs more sufficient conditions, which may be impossible to satisfy theoretically, than that of Ritz values does The issues of how to restart and to solve multiple eigenproblems are addressed, and some numerical examples are reported to confirm the theoretical analysis