Showing papers in "Numerische Mathematik in 1990"
TL;DR: In this paper, the authors developed discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume.
Abstract: The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH 1 semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h 2). Results on the effects of numerical integration are also included.
353 citations
TL;DR: In this paper, an algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of Poisson problems on a set of surfaces.
Abstract: An Algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of Poisson problems on a series of surfaces. This gives a new method to solve Plateau's problem forH-surfaces.
352 citations
TL;DR: In this article, a convergence and error analysis for a Nystrom method on a graded mesh for the numerical solution of boundary integral equations for the harmonic Dirichlet problem in plane domains with corners is given.
Abstract: We give a convergence and error analysis for a Nystrom method on a graded mesh for the numerical solution of boundary integral equations for the harmonic Dirichlet problem in plane domains with corners.
212 citations
TL;DR: In this article, a class of regularization methods for a discretized version of an operator equation with approximately given right-hand side is considered, including the case that the problem is ill-posed.
Abstract: In this paper we consider a class of regularization methods for a discretized version of an operator equation (which includes the case that the problem is ill-posed) with approximately given right-hand side. We propose an a priori- as well as an a posteriori parameter choice method which is similar to the discrepancy principle of Ivanov-Morozov. From results on fractional powers of selfadjoint operators we obtain convergence rates, which are (in many cases) the same for both parameter choices.
151 citations
TL;DR: The Runge-Kutta-Chebyshev method as mentioned in this paper is an ans-stage runge-kutta method designed for the explicit integration of stiff systems of ordinary differential equations originating from spatial discretization of parabolic partial differential equations.
Abstract: The Runge-Kutta-Chebyshev method is ans-stage Runge-Kutta method designed for the explicit integration of stiff systems of ordinary differential equations originating from spatial discretization of parabolic partial differential equations (method of lines). The method possesses an extended real stability interval with a length β proportional tos 2. The method can be applied withs arbitrarily large, which is an attractive feature due to the proportionality of β withs 2. The involved stability property here is internal stability. Internal stability has to do with the propagation of errors over the stages within one single integration step. This internal stability property plays an important role in our examination of full convergence properties of a class of 1st and 2nd order schemes. Full convergence means convergence of the fully discrete solution to the solution of the partial differential equation upon simultaneous space-time grid refinement. For a model class of linear problems we prove convergence under the sole condition that the necessary time-step restriction for stability is satisfied. These error bounds are valid for anys and independent of the stiffness of the problem. Numerical examples are given to illustrate the theoretical results.
135 citations
TL;DR: The hierarchical basis preconditioner of Bramble, Pasciak and Xu are derived and analyzed within a joint framework and the close relationship between both methods is elucidated.
Abstract: The hierarchical basis preconditioner and the recent preconditioner of Bramble, Pasciak and Xu are derived and analyzed within a joint framework. This discussion elucidates the close relationship between both methods. Special care is devoted to highly nonuniform meshes; exclusively local properties like the shape regularity of the finite elements are utilized.
134 citations
TL;DR: The algorithm, called the nonlinear Galerkin method, stems from the theory of dynamical systems and amounts to some approximation of the attractor in the discrete (finite elements) space which is accomplished in finite elements by using hierarchical bases.
Abstract: With the increase in the computing power and the advent of supercomputers, the approximation of evolution equations on large intervals of time is emerging as a new type of numerical problem. In this article we consider the approximation of evolution equations on large intervals of time when the space discretization is accomplished by finite elements. The algorithm that we propose, called the nonlinear Galerkin method, stems from the theory of dynamical systems and amounts to some approximation of the attractor in the discrete (finite elements) space. Essential here is the utilization of incremental unknown which is accomplished in finite elements by using hierarchical bases. Beside a detailed description of the algorithm, the article includes some technical results on finite elements spaces, and a full study of the stability and convergence of the method.
121 citations
TL;DR: In this article, the authors study the convergence of iterative methods for algebraic linear systems of equations and present conditions on the splittings corresponding to the iterative method to guarantee convergence for any number of inner iterations.
Abstract: Classical iterative methods for the solution of algebraic linear systems of equations proceed by solving at each step a simpler system of equations. When this system is itself solved by an (inner) iterative method, the global method is called a two-stage iterative method. If this process is repeated, then the resulting method is called a nested iterative method. We study the convergence of such methods and present conditions on the splittings corresponding to the iterative methods to guarantee convergence forany number of inner iterations. We also show that under the conditions presented, the spectral radii of the global iteration matrices decrease when the number of inner iterations increases. The proof uses a new comparison theorem for weak regular splittings. We extend our results to larger classes of iterative methods, which include iterative block Gauss-Seidel. We develop a theory for the concatenation of such iterative methods. This concatenation appears when different numbers of inner interations are performed at each outer step. We also analyze block methods, where different numbers of inner iterations are performed for different diagonal blocks.
107 citations
TL;DR: In this paper, abstract error estimates for the approximation of the velocity and the pressure by a mixed FEM of quasi-Newtonian flows whose viscosity obeys the power law or the Carreau law are presented.
Abstract: We prove abstract error estimates for the approximation of the velocity and the pressure by a mixed FEM of quasi-Newtonian flows whose viscosity obeys the power law or the Carreau law These estimates are optimal in some cases They can be applied to most finite elements used for the solution of Stokes's problem
On prouve des estimations d'erreur abstraites pour l'approximation de la vitesse et la pression par une MEF mixtes d'ecoulements quasi-Newtoniens dont la viscosite obeit a la loi puissance ou la loi de Carreau Ces estimations sont optimales dans certains cas Elles peuvent etre appliquees a la plupart des elements finis utilises pour la resolution du probleme de Stokes
100 citations
TL;DR: Three different conjugate gradient type approaches with iterates defined by a minimal residual property, a Galerkin type condition, and an Euclidean error minimization are investigated and numerical experiments for matrices arising from finite difference approximations to the complex Helmholtz equation are reported on.
Abstract: We consider conjugate gradient type methods for the solution of large linear systemsA x=b with complex coefficient matrices of the typeA=T+i?I whereT is Hermitian and ? a real scalar. Three different conjugate gradient type approaches with iterates defined by a minimal residual property, a Galerkin type condition, and an Euclidean error minimization, respectively, are investigated. In particular, we propose numerically stable implementations based on the ideas behind Paige and Saunder's SYMMLQ and MINRES for real symmetric matrices and derive error bounds for all three methods. It is shown how the special shift structure ofA can be preserved by using polynomial preconditioning, and results on the optimal choice of the polynomial preconditioner are given. Also, we report on some numerical experiments for matrices arising from finite difference approximations to the complex Helmholtz equation.
90 citations
TL;DR: In this article, the comparison theorem for weak splittings of bounded operators is extended for general Banach spaces and rather general cones, and strong inequalities for the comparisons are given.
Abstract: Comparison theorems for weak splittings of bounded operators are presented. These theorems extend the classical comparison theorem for regular splittings of matrices by Varga, the less known result by Wo?nicki, and the recent results for regular and weak regular splittings of matrices by Neumann and Plemmons, Elsner, and Lanzkron, Rose and Szyld. The hypotheses of the theorems presented here are weaker and the theorems hold for general Banach spaces and rather general cones. Hypotheses are given which provide strict inequalities for the comparisons. It is also shown that the comparison theorem by Alefeld and Volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators.
TL;DR: In this paper, it was shown that the distance inL 2 between the approximate solution and a projection of the exact one is of higher order than the error itself, and this result is exploited to obtain superconvergence at Gaussian points and to construct higher order approximations by a local postprocessing.
Abstract: In this paper we prove superconvergence error estimates for the vector variable for mixed finite element approximations of second order elliptic problems. For the rectangular finite elements of Raviart and Thomas [19] and for those of Brezzi et al. [4] we prove that the distance inL 2 between the approximate solution and a projection of the exact one is of higher order than the error itself.
This result is exploited to obtain superconvergence at Gaussian points and to construct higher order approximations by a local postprocessing.
TL;DR: The approach is used to examine several upwinding schemes, including the streamline diffusion method, the boxmethod, the Scharfetter-Gummel discretization, and a divergence-free scheme.
Abstract: A uniform framework for the study of upwinding schemes is developed. The standard finite element Galerkin discretization is chosen as the reference discretization, and differences between other discretization schemes and the reference are written as artificial diffusion terms. These artificial diffusion terms are spanned by a four dimensional space of element diffusion matrices. Three basis matrices are symmetric, rank one diffusion operators associated with the edges of the triangle; the fourth basis matrix is skew symmetric and is associated with a rotation by ?/2. While finite volume discretizations may be written as upwinded Galerkin methods, the converse does not appear to be true. Our approach is used to examine several upwinding schemes, including the streamline diffusion method, the box method, the Scharfetter-Gummel discretization, and a divergence-free scheme.
TL;DR: In this article, a convergence theory for multi-grid methods with transforming smoothers is presented for a general system of partial differential equations, which follows Hackbusch's approach for scalar pde and allows a convergence proof for some well-known multigrid methods for Stokes and Navier-Stokes equations as DGS by Brandt-Dinar, [5], TILU from [31] and the SIMPLE-methods by Patankar-Spalding, [23].
Abstract: In the present paper we give a convergence theory for multi-grid methods with transforming smoothers as introduced in [31] applied to a general system of partial differential equations. The theory follows Hackbusch's approach for scalar pde and allows a convergence proof for some well-known multi-grid methods for Stokes- and Navier-Stokes equations as DGS by Brandt-Dinar, [5], TILU from [31] and the SIMPLE-methods by Patankar-Spalding, [23].
TL;DR: How this eigenproblem can be solved by a divide and conquer method, in which the matrixH is split into two smaller unitary upper Hessenberg matricesH1 andH2 by a rank-one modification ofH, is described.
Abstract: LetH?? n xn be a unitary upper Hessenberg matrix whose eigenvalues, and possibly also eigenvectors, are to be determined. We describe how this eigenproblem can be solved by a divide and conquer method, in which the matrixH is split into two smaller unitary upper Hessenberg matricesH 1 andH 2 by a rank-one modification ofH. The eigenproblems forH 1 andH 2 can be solved independently, and the solutions of these smaller eigenproblems define a rational function, whose zeros on the unit circle are the eigenvalues ofH. The eigenvector ofH can be determined from the eigenvalues ofH and the eigenvectors ofH 1 andH 2. The outlined splitting of unitary upper Hessenberg matrices into smaller such matrices is carried out recursively. This gives rise to a divide and conquer method that is suitable for implementation on a parallel computer.
WhenH?? n xn is orthogonal, the divide and conquer scheme simplifies and is described separately. Our interest in the orthogonal eigenproblem stems from applications in signal processing. Numerical examples for the orthogonal eigenproblem conclude the paper.
TL;DR: In this article, the authors considered the well-known spaces of bivariate piecewise polynomials of degreed defined over arbitrary triangulations of a polygonal domain and possessing r continuous derivatives globally.
Abstract: We consider the well-known spaces of bivariate piecewise polynomials of degreed defined over arbitrary triangulations of a polygonal domain and possessingr continuous derivatives globally. To date, dimension formulae for such spaces have been established only whend?3r+2, (except for the special case wherer=1 andd=4). In this paper we establish dimension formulae for allr?1 andd=3r+1 for almost all triangulations.
TL;DR: In this article, a particle method based on a viscous splitting of the operator was proposed to solve the Navier-Stokes equations. But the method was not considered in this paper.
Abstract: We extend to the case of the two-dimensional Navier-Stokes equations, a particle method introduced in a previous paper to solve linear convection-diffusion equations. The method is based on a viscous splitting of the operator. The particles move under the effect of the velocity field but are not affected by the diffusion which is taken into account by the weights. We prove the stability and the convergence of the method.
TL;DR: In this paper, the precise space of functions for which the trapezoidal rule converges with a prescribed rate as the number of nodes tends to infinity is determined, and a correspondence between the speed of convergence and regularity properties of the function is established.
Abstract: Our aim is to determine the precise space of functions for which the trapezoidal rule converges with a prescribed rate as the number of nodes tends to infinity. Excluding or controlling odd functions in some way it is possible to establish a correspondence between the speed of convergence and regularity properties of the function to be integrated. In this way we characterize Sobolev spaces, certain spaces of infinitely differentiable functions, of functions holomorphic in a strip, of entire functions of order greater than 1 and of entire functions of exponential type by the speed of convergence.
TL;DR: In this article, a method to generate an accurate approximation to a singular solution of a system of complex analytic equations is presented, using homotopy continuation and 1-dimensional quadrature.
Abstract: A method to generate an accurate approximation to a singular solution of a system of complex analytic equations is presented. Since manyreal systems extend naturally tocomplex analytic systems, this porvides a method for generating approximations to singular solutions to real systems. Examples include systems of polynomials and systems made up of trigonometric, exponential, and polynomial terms. The theorem on which the method is based is proven using results from several complex variables. No special conditions on the derivatives of the system, such as restrictions on the rank of the Jacobian matrix at the solution, are required. The numerical method itself is developed from techniques of homotopy continuation and 1-dimensional quadrature. A specific implementation is given, and the results of numerical experiments in solving five test problems are presented.
TL;DR: In this article, an active set approach is used to determine appropriate active sets from solutions to linear complementarity problems, and convergence results are established under some non-degeneracy assumptions.
Abstract: Ordinary Differential Equations with discontinuities in the state variables require a differential inclusion formulation to guarantee existence [8]. From this formulation a high accuracy method for solving such initial value problems is developed which can give any order of accuracy and "tracks" the discontinuities. The method uses an "active set" approach, and determines appropriate active sets from solutions to Linear Complementarity Problems. Convergence results are established under some non-degeneracy assumptions. The method has been implemented, and results compare favourably with previously published methods [7, 21].
TL;DR: In this paper, the condition number of the Schur complement is shown to be smaller than condition number obtained by the block-diagonal preconditioning, where the first block of variables consists of degrees of freedom of a low order.
Abstract: We study symmetric positive definite linear systems, with a 2-by-2 block matrix preconditioned by inverting directly one of the diagonal blocks and suitably preconditioning the other. Using an approximate version of Young's "Property A", we show that the condition number of the Schur complement is smaller than the condition number obtained by the block-diagonal preconditioning. We also get bounds on both condition numbers from a strengthened Cauchy inequality. For systems arising from the finite element method, the bounds do not depend on the number of elements and can be obtained from element-by-element computations. The results are applied to thep-version finite element method, where the first block of variables consists of degrees of freedom of a low order.
TL;DR: In this paper, a nonlinear evolutionary system of partial differential equations arising from modeling the dynamics of solid-state phase transitions of thermomechenical nature in the case of one space dimension is constructed.
Abstract: Discrete approximations are constructed to a nonlinear evolutionary system of partial differential equations arising from modelling the dynamics of solid-state phase transitions of thermomechenical nature in the case of one space dimension. The class of problems considered includes the so-called shape memory alloys, in particular. It is shown that the obtained discrete solutions converge to the solution of the original problem, and numerical simulations for the shape memory alloy Au23Cu30Zn47 demonstrate the quality of the discrete model.
TL;DR: In this article, it is shown that even though these methods converge in the case of exact data their instability makes it impossible to base a-priori parameter choice regularization methods upon them.
Abstract: For the solution of linear ill-posed problems some gradient methods like conjugate gradients and steepest descent have been examined previously in the literature. It is shown that even though these methods converge in the case of exact data their instability makes it impossible to base a-priori parameter choice regularization methods upon them.
TL;DR: In this paper, the authors studied the nonlinear stability and the convergence of some simple finite-difference schemes for the numerical solution of problems involving the good Boussinesq equation.
Abstract: The "good" Boussinesq equationu tt =?u xxxx +u xx +(u 2) xx has recently been found to possess an interesting soliton-interaction mechanism. In this paper we study the nonlinear stability and the convergence of some simple finite-difference schemes for the numerical solution of problems involving the "good" Boussinesq equation. Numerical experimentas are also reported.
TL;DR: In this paper, the authors studied the finite element approximation of nonlinear second order elliptic boundary value problems with discontinuous coefficients with mixed Dirichlet-Neumann boundary conditions.
Abstract: The study of the finite element approximation to nonlinear second order elliptic boundary value problems with discontinuous coefficients is presented in the case of mixed Dirichlet-Neumann boundary conditions. The change in domain and numerical integration are taken into account. With the assumptions which guarantee that the corresponding operator is strongly monotone and Lipschitz-continuous the following convergence results are proved: 1. the rate of convergenceO(h ?) if the exact solutionu?H 1 (Ω) is piecewise of classH 1+? (0
TL;DR: In this article, a generalization of interpolation called μ-interpolation and a notion of higher order quasi interpolation are introduced, which allows for their direct calculation in a variety of forms suitable to particular applications, and in addition, provides a clear formulation of the difficult problem of minimally supported quasiinterpolants.
Abstract: Let ? be a compactly supported function on ? s andS (?) the linear space withgenerator ?; that is,S (?) is the linear span of the multiinteger translates of ?. It is well known that corresponding to a generator ? there are infinitely many quasi-interpolation formulas. A characterization of these formulas is presented which allows for their direct calculation in a variety of forms suitable to particular applications, and in addition, provides a clear formulation of the difficult problem of minimally supported quasi-interpolants. We introduce a generalization of interpolation called μ-interpolation and a notion of higher order quasi-interpolation called μ-approximation. A characterization of μ-approximants similar to that of quasi-interpolants is obtained with similar applications. Among these applications are estimating least-squares approximants without matrix inversion, surface fitting to incomplete or semi-scattered discrete data, and constructing generators with one-point quasi-interpolation formulas. It will be seen that the exact values of the generator ? at the multi-integers ? s facilitates the above study. An algorithm to yield this information for box splines is discussed.
TL;DR: In this paper, a numerical method that detects singular minimizers and avoids the Lavrentiev phenomenon for three dimensional problems in nonlinear elasticity is described and analyzed. And the method extends to three dimensions the corresponding one dimensional method of Ball and Knowles.
Abstract: In this paper we describe and analyse a numerical method that detects singular minimizers and avoids the Lavrentiev phenomenon for three dimensional problems in nonlinear elasticity. This method extends to three dimensions the corresponding one dimensional method of Ball and Knowles.
TL;DR: In this article, the authors provided a recursive test to check whether a regular Runge-Kutta method is regular and proved that the order of an ans-stage regular method may not exceed 2[(s+2)/2] and that the maximal order of regular runge-kutta methods with an irreducible stability function is 4.
Abstract: It is known that certain Runge-Kutta methods share the property that, in a constant-step implementation, if a solution trajectory converges to a bounded limit then it must be a fixed point of the underlying differential system. Such methods are calledregular. In the present paper we provide a recursive test to check whether given method is regular. Moreover, by examining solution trajectories of linear equations, we prove that the order of ans-stage regular method may not exceed 2[(s+2)/2] and that the maximal order of regular Runge-Kutta method with an irreducible stability function is 4.
TL;DR: In this paper, the order of the difference between the numerical approximation and the solution of a parabolic variational inequality is estimated using a finite element discretization in space and a finite difference discretisation in time which is more general than is used in the literature.
Abstract: We estimate the order of the difference between the numerical approximation and the solution of a parabolic variational inequality. The numerical approximation is obtained using a finite element discretization in space and a finite difference discretization in time which is more general than is used in the literature. We obtain better error estimates than those given in the literature. The error estimates are compared with numerical experiments.
TL;DR: In this paper, the singular values of Vandermonde matrices and their spectral condition numbers are studied. But the main interest is in the singular value of these matrices, not the spectral condition number.
Abstract: We study Vandermonde matrices whose nodes are given by a Van der Corput sequence on the unit circle. Our primary interest is in the singular values of these matrices and the respective (spectral) condition numbers. Detailed information about multiplicities and eigenvectors, however, is also obtained. Two applications are given to the theory of polynomials.