A finite difference discretization method for elliptic problems on composite grids

In this paper we discusss a simple finite difference method for the discretization of elliptic boundary value problems on composite grids. For the model problem of the Poisson equation we prove stability of the discrete operator and bounds for the global discretization error. These bounds clearly show how the discretization error depends on the grid size of the coarse grid, on the grid size of the local fine grid and on the order of the interpolation used on the interface. Furthermore, the constants in these bounds do not depend on the quotient of coarse grid size and fine grid size. We also discuss an efficient solution method for the resulting composite grid algebraic problem.     

Computation of QFT bounds for robust tracking specifications

An algorithm is proposed for generation of QFT controller bounds to achieve robust tracking specifications. The proposed algorithm uses quadratic constraints and interval plant templates to compute the bounds, and presents several improvements over existing QFT tracking bound generation algorithms. The proposed algorithm (1) guarantees robustness against template inaccuracies, (2) guarantees robustness against phase discretization, (3) provides a posteriori error estimates, (4) is computationally efficient, achieving a reduction in flops and execution time, typically by 1-2 orders of magnitude. The algorithm is demonstrated on an aircraft example having five uncertain parameters.     

Further analysis of the local defect correction method

We analyze a special case of the Local Defect Correction (LDC) method introduced in [4]. We restrict ourselves to finite difference discretizations of elliptic boundary value problems. The LDC method uses the discretization on a uniform global coarse grid and on one or more uniform local fine grids for approximating the continuous solution. We prove that this LDC method can be seen as an iterative method for solving an underlying composite grid discretization. This result makes it possible to explain important properties of the LDC method, e.g. concerning the size of the discretization error. Furthermore, the formulation of LDC as an iterative solver for a given composite grid problem makes it possible to prove a close correspondence between LDC and the Fast Adaptive Composite grid (FAC) method from [8–10].     

A Posteriori Error Estimators for a Class of Variational Inequalities

In this paper, we present an a posteriori error analysis for the finite element approximation of a variational inequality. We derive a posteriori error estimators of residual type, which are shown to provide upper bounds on the discretization error for a class of variational inequalities provided the solutions are sufficiently regular. Furthermore we derive sharp a posteriori error estimators with both lower and upper error bounds for a subclass of the obstacle problem which are frequently met in many physical models. For sufficiently regular solutions, these estimates are shown to be equivalent to the discretization error in an energy type norm. Our numerical tests show that these sharp error estimators are both reliable and efficient in guiding mesh adaptivity for computing the free boundaries.     

Error estimation in the integration of ordinary differential equations

Linear initial value problems, particularly involving first order differential equations, can be transformed into systems of higher order and treated as boundary value problems. Finite difference analogues considered for obtaining approximate solutions of these boundary value problems are proved to be fourth order convergent processes, by deriving considerable sharper bounds for the discretization error. Numerical examples are given to demonstrate the usefulness of our error bounds.     

Stability of Galerkin multistep procedures in time-homogeneous hyperbolic problems

Linear and time-homogeneous hyperbolic initial boundary value problems are approximated using Galerkin procedures for the space directions and linear multistep methods for the time direction. At first error bounds are proved for multistep methods having a stability interval [?ω, 0], 0<ω, and systemsY″=AY+C(t) under the condition that \(\Delta t^2 \left\| A \right\| \leqslant \omega \) Δt time step. Then these error bounds are applied to derive bounds for the error in hyperbolic problems. The result shows that the initial error and the discretization error grow liket andt ² respectively. But the initial error is multiplied with a factor which becomes large if the mesh width of the space discretization is small.     

A weighted residual relationship for the contact problem with Coulomb friction

In this work, a weighted residual relationship is proposed as an extension of the standard virtual work principle to deal with the large deformation contact problem with Coulomb friction. This weak form is a mixed relationship involving the displacements and the multipliers defined on the reference contact surface of the contactor and is shown to be equivalent to the strong form of the initial/boundary value contact problem. The discretization in space by means of the finite element method is carried out on the mixed relationship in a simple way in order to obtain the semi-discrete equation system. The contact tangent stiffness is derived and numerical examples are presented to assess the efficiency of the formulation.     

Vorticity–velocity–pressure formulation for Navier–Stokes equations

We analyze here the bidimensional boundary value problems, for both Stokes and Navier–Stokes equations, in the case where non standard boundary conditions are imposed. A well-posed vorticity–velocity–pressure formulation for the Stokes problem is introduced and its finite element discretization, which needs some stabilization, is then studied. We consider next the approximation of the Navier–Stokes equations, based on the previous approximation of the Stokes equations. For both problems, the convergence of the numerical approximation and optimal error estimates are obtained. Some numerical tests are also presented.     

Spectral element discretization of the vorticity, velocity and pressure formulation of the Navier–Stokes problem

Abstract The two-dimensional Navier–Stokes equations, when subject to non-standard boundary conditions which involve the normal component of the velocity and the vorticity, admit a variational formulation with three independent unknowns, the vorticity, velocity and pressure. We propose a discretization of this problem by spectral element methods. A detailed numerical analysis leads to optimal error estimates for the three unknowns and numerical experiments confirm the interest of the discretization.     

Characterizing and approximating eigenvalue sets of symmetric interval matrices

We consider the eigenvalue problem for the case where the input matrix is symmetric and its entries are perturbed, with perturbations belonging to some given intervals. We present a characterization of some of the exact boundary points, which allows us to introduce an inner approximation algorithm, that in many case estimates exact bounds. To our knowledge, this is the first algorithm that is able to guarantee exactness. We illustrate our approach by several examples and numerical experiments.     

Automatic Forward Error Analysis for Floating Point Algorithms

We investigate absolute and relative error bounds for floating point calculations determined by means of sequences of instructions (as, for example, given by a computer program). We get rigorous error bounds on the round-off or generated error due to the actual machine floating point operations, as well as the propagated error from one sequence to the next in a very convenient way by the computer itself. The results stated in the theorems can be used to implement software tools for the automatic computation of a priori worst case error bounds for floating point computations. These automatically computed bounds are valid simultaneously for all data vectors varying in the domain specified and their corresponding machine vectors fulfilling a maximum prescribed error bound.With great success we have used our method in the past to implement a fast interval library for elementary functions called FI_LIB [12]. Further numerical examples often show a high quality of the computed a priori bounds.     

Worst case identification of continuous time systems via interpolation

We consider a worst case robust control oriented identification problem recently studied by several authors. This problem is one of identification in the continuous time setting. We give a more general formulation of this problem. The available a priori information in this paper consists of a lower bound on the relative stability of the plant, a frequency dependent upper bound on a certain gain associated with the plant, and an upper bound on the noise level. The available experimental information consists of a finite number of noisy plant point frequency response samples. The objective is to identify, from the given a priori and experimental information, an uncertain model that includes a stable nominal plant model and a bound on the modeling error measured in norm. Our main contributions include both a new identification algorithm and several new ‘explicit’ lower and upper bounds on the identification error. The proposed algorithm belongs to the class of ‘interpolatory algorithms’ which are known to possess a desirable optimality property under a certain criterion. The error bounds presented improve upon the previously available ones in the aspects of both providing a more accurate estimate of the identification error as well as establishing a faster convergence rate for the proposed algorithm.     

The elastic problem for fractal media: basic theory and finite element formulation

In a previous paper [Comput. Methods Appl. Mech. Eng. 190 (2001) 6053], the framework for the mechanics of solids, deformable over fractal subsets, was outlined. Anomalous mechanical quantities with fractal dimensions were introduced, i.e., the fractal stress [σ∗], the fractal strain [ε∗] and the fractal work of deformation W∗. By means of the local fractional operators, the static and kinematic equations were obtained, and the Principle of Virtual Work for fractal media was demonstrated. In this paper, the constitutive equations of fractal elasticity are put forward. From the definition of the fractal elastic potential φ∗, the linear elastic constitutive relation is derived. The physical dimensions of the second derivatives of the elastic potential depend on the fractal dimensions of both stress and strain. Thereby, the elastic constants undergo positive or negative scaling, depending on the topological character of deformation patterns and stress flux. The direct formulation of elastic equilibrium is derived in terms of the fractional Lamé operators and of the equivalence equations at the boundary. The variational form of the elastic problem is also obtained, through minimization of the total potential energy. Finally, discretization of the fractal medium is proposed, in the spirit of the Ritz-Galerkin approach, and a finite element formulation is obtained by means of devil’s staircase interpolating splines.     

Mixed finite-difference scheme for analysis of simply supported thick plates

A mixed finite-difference scheme is presented for the stress and free vibration analysis of simply supported nonhomogeneous and layered orthotropic thick plates. The analytical formulation is based on the linear, three-dimensional theory of orthotropic elasticity and a Fourier approach is used to reduce the governing equations to six first-order ordinary differential equations in the thickness coordinate. The governing equations possess a symmetric coefficient matrix and are free of derivatives of the elastic characteristics of the plate. In the finite difference discretization two interlacing grids are used for the different fundamental unknowns in such a way as to reduce both the local discretization error and the bandwidth of the resulting finite-difference field equations. Numerical studies are presented for the effects of reducing the interior and boundary discretization errors and of mesh refinement on the accuracy and convergence of solutions. It is shown that the proposed scheme, in addition to a number of other advantages, leads to highly accurate results, even when a small number of finite difference intervals is used.     

Computable explicit bounds for the discretization error of variable stepsize multistep methods

This paper deals with the achievement of explicit computable bounds for the global discretization error of variable stepsize multistep methods which are perturbation of strongly stable fixed stepsize methods. The approach is based on the study of the growth of solutions of certain variable coefficient difference equations satisfied by the global discretization error.     

Solution of mixed boundary value problems with local error bound by the finite element method

A method of analysis using finite element techniques is presented for second order, mixed boundary value problems in the plane. The technique focuses computational effort on specific points in the domain and provides absolute solution error bounds at those points by applying the hypercircle method. Solution error of less than 0.0003% and solution error bounds of ± 0.012% are obtained in sample problems. The solution accuracy is notably superior to what is obtained in the traditional finite element method with equivalent discretization. Two problems are presented to illustrate both the strengths and weaknesses of the method.     

Direct boundary integral equation method for electromagnetic scattering by partly coated dielectric objects

We present a new variational direct boundary integral equation approach for solving the scattering and transmission problem for dielectric objects partially coated with a PEC layer. The main idea is to use the electromagnetic Calderón projector along with transmission conditions for the electromagnetic fields. This leads to a symmetric variational formulation which lends itself to Galerkin discretization by means of divergence-conforming discrete surface currents. A wide array of numerical experiments confirms the efficacy of the new method. Dedicated to George C. Hsiao on the occasion of his 70th birthday. Communicated by: W. L. Wendland     

A Computer Oriented Approach to Get Sharp Reliable Error Bounds

We present interval methods to get reliable a priori error bounds for the machine evaluation of algorithms implementing some mathematical expression. The term expression not only means simple arithmetical expressions but also more complex program parts including loops or recursive structures (e.g. a complete elementary function routine).We sketch a method that can be used to get an upper bound for the approximation error of a polynomial or a rational approximation. We also discuss a method to compute worst case a priori error estimates for arbitrary IEEE double floating-point computations. Our theoretical results lead to reliable and easy to use public domain software tools. The application of these tools to an accurate table method shows that error bounds of high quality can be derived.     

A Spectral Mortar Element Discretization of the Poisson Equation with Mixed Boundary Conditions

In this paper, we study a spectral mortar element discretization of the Poisson equation on a square subject to mixed boundary conditions of Dirichlet and Neumann type. We carry out the numerical analysis of the method and derive error estimates. An efficient algorithm for the solution of the problem is proposed and numerical tests confirming the theoretical results are presented.     

A posteriori error estimates for fem–bem couplings of three-dimensional electromagnetic problems

We present residual based and p-hierarchical a posteriori error estimators for a Galerkin method coupling finite elements and boundary elements for time–harmonic interface problems in electromagnetics; special emphasis is taken for the eddy current problem. The Galerkin discretization uses lowest order Nédélec elements in the interior domain and vectorial surface rotations of continuous, piecewise linear functions on the interface boundary. Singular, weakly singular and hypersingular boundary integral operators appearing in the variational formulation show up in the terms of the error estimators as well. The estimators are derived from the defect equation using Helmholtz and Hodge decompositions. Numerical tests underline reliability and efficiency of the given error estimators yielding reasonable mesh refinements.