Local and global accuracy estimates for boundary element analysis

An adaptive version of Glimm's scheme

In this study, we give an a posteriori error analysis on the weighted essentially nonoscillatory schemes for the nonlinear scalar conservation laws. This analysis is based on the new concept of numerical smoothness, with some new error analysis mechanisms developed for the finite difference and finite volume discretizations. The local error estimate is of optimal order in space and time. The global error estimate grows linearly in time, because of the direct application of the L1 ‐contraction between entropy solutions in the error propagation analysis. As a beginning, we only deal with smooth solutions in this article. Within the same error propagation framework, when we deal with piecewise smooth solutions later, we only need to work on estimating the local error where smoothness is lost. The smoothness indicators not only serve the purpose of local error estimation, but also serve as a monitor on both the possible numerical instability and the expected solution shapening. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

Collections of linear or nonlinear operator equations Au = f are considered which may represent (i) differential or integral equations or (ii) finite-dimensional approximations. Input sets of coefficients a or data f are admitted. The envelope of the set of solutions is to be constructed where this boundary refers (i) to the ränge of values of the solutions or (ii) to a finite-dimensional space. The construction employs either topological boundary mapping or truncated Taylor expansions. Estimates of the local procedural errors are due to suitable a priori sets and interval mathematics. The relation between local and global error estimates is due to boundary mapping or an auxiliary inverse-monotone operator B. The operator B is constructed for the case of arbitrary linear ordinary differential equations with boundary or initial conditions, provided the admitted A satisfy a mild condition.KeywordsBoundary MappingOperator EquationRemainder TermDiscrete AnalogyOuter ApproximationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper discusses the numerical approximation of stable dynamical systems of ordinary differential equations by general time-stepping methods. The traditional error analysis for this approximation yields estimates for the global discretization error in terms of the local truncations errors with constants generally growing exponentially in time due to the use of a discrete Gronwall inequality. In special cases of monotone systems, global error estimates can be shown to hold uniformly in time provided that also the time stepping scheme possesses certain monotonicity properties. The standard example is the backward Euler method. However, for more general time-stepping schemes, this is not clear even in the monotone case. It is shown here how in general situations certain qualitative stability properties (exponential and quasi-exponential stability) of the solution to be approximated can be used for extending local error estimates to global ones holding uniformly in time. Further, the approximate solutions are likewise stable and, in the autonomous case, tend to equilibrium points.

The approximate dynamics of many physical phenomena, including turbulence, can be represented by dissipative systems of ordinary differential equations. One often turns to numerical integration to solve them. There is an incompatibility, however, between the answers it can produce (i.e., specific solution trajectories) and the questions one might wish to ask (e.g., what behavior would be typical in the laboratory?) To determine its outcome, numerical integration requires more detailed initial conditions than a laboratory could normally provide. In place of initial conditions, experiments stipulate how tests should be carried out: only under statistically stationary conditions, for example, or only during asymptotic approach to a final state. Stipulations such as these, rather than initial conditions, are what determine outcomes in the laboratory. This theoretical study examines whether the points of view can be reconciled: What is the relationship between one's statistical stipulations for how an experiment should be carried out--stationarity or asymptotic approach--and the expected results? How might those results be determined without invoking initial conditions explicitly? To answer these questions, stationarity and asymptotic approach conditions are analyzed in detail. Each condition is treated as a statistical constraint on the system--a restriction on the probability density of states that might be occupied when measurements take place. For stationarity, this reasoning leads to a singular, invariant probability density which is already familiar from dynamical systems theory. For asymptotic approach, it leads to a new, more regular probability density field. A conjecture regarding what appears to be a limit relationship between the two densities is presented. By making use of the new probability densities, one can derive output statistics directly, avoiding the need to create or manipulate initial data, and thereby avoiding the conceptual incompatibility mentioned above. This approach also provides a clean way to derive reduced-order models, complete with local and global error estimates, as well as a way to compare existing reduced-order models objectively. The new approach is explored in the context of five separate test problems: a trivial one-dimensional linear system, a damped unforced linear oscillator in two dimensions, the isothermal Rayleigh-Plesset equation, Lorenz's equations, and the Stokes limit of Burgers' equation in one space dimension. In each case, various output statistics are deduced without recourse to initial conditions. Further, reduced-order models are constructed for asymptotic approach of the damped unforced linear oscillator, the isothermal Rayleigh-Plesset system, and Lorenz's equations, and for stationarity of Lorenz's equations.

Recently a refined approach to error control in finite element (FE) discretisations has been proposed, Becker and Rannacher (1995b), (1996), which uses weighted a posteriori error estimates derived via duality arguments. The conventional strategies for mesh refinement in FE models of problems from elasticity theory are mostly based on a posteriori error estimates in the energy norm. Such estimates reflect the approximation properties of the finite element ansatz by local interpolation constants while the stability properties of the continuous model enter through a global coercivity constant. However, meshes generated on the basis of such global error estimates are not appropriate in cases where the domain consists of very heterogeneous materials and for the computation of local quantities, e.g., point values or contour integrals. This deficiency is cured by using certain local norms of the dual solution directly as weights multiplying the local residuals of the computed solution. In general, these weights have to be evaluated numerically in the course of the refinement process, yielding almost optimal meshes for various kinds of error measures. This feed-back approach is developed here for primal as well as mixed FE discretisations of the fundamental problem in linear elasticity.

By means of the 2-stage ROW methods, applied to a stiff model, Stetter's global error estimation is studied in the smooth phase. It turns out that this error estimation, which is based on theh→0 asymptotic, remains also reliable forh≫0 if the parameters of the method are chosen in a certain way. The relations between the effectiveness of the error estimation, the stiffness of the problem and the parameters of the method are investigated by means of an error expansion, the coefficients of which are bounded for decreasingh as well as for increasing stiffness.

In this work, we develop an efficient incomplete iterative scheme for the numerical solution of the subdiffusion model involving a Caputo derivative of order $$\alpha \in (0,1)$$ in time. It is based on piecewise linear Galerkin finite element method in space and backward Euler convolution quadrature in time and solves one linear algebraic system inexactly by an iterative algorithm at each time step. We present theoretical results for both smooth and nonsmooth solutions, using novel weighted estimates of the time-stepping scheme. The analysis indicates that with the number of iterations at each time level chosen properly, the error estimates are nearly identical with that for the exact linear solver, and the theoretical findings provide guidelines on the choice. Illustrative numerical results are presented to complement the theoretical analysis.

In recent papers the technique for a local and global error estimation and the local-global step size control were presented to solve both ordinary differential equations and semi-explicit index 1 differential-algebraic systems by multistep methods with any reasonable accuracy attained automatically. Now those results are extended to the concept of multistep extrapolation, and the paper demonstrates with numerical examples how such methods work in practice. Especially, we develop an efficient technique to calculate higher derivatives of a numerical solution with Hermite interpolating polynomials. The necessary theory is also provided.

The paper consists of two parts. In the first part, we propose a procedure to estimate local errors of low order methods applied to solve initial value problems in ordinary differential equations (ODEs) and index 1 differential-algebraic equations (DAEs). Based on the idea of defect correction we develop local error estimates for the case when the problem data is only moderately smooth. Numerical experiments illustrate the performance of the mesh adaptation based on the error estimation developed in this paper. In the second part of the paper, we will consider the estimation of local errors in context of stochastic differential equations with small noise.

This paper proposes a new methodology for robotic offline programming (OLP) addressing the issue of automatic program generation directly from 3D CAD models and verification through online 3D reconstruction. Limitations of current OLP include manufacturing tolerances between CAD and workpieces and inaccuracies in workpiece placement and modelled work cell. These issues are addressed and demonstrated through surface scanning, registration, and global and local error estimation. The method allows the robot to adjust the welding path designed from the CAD model to the actual workpiece. Alternatively, for non-repetitive tasks and where a CAD model is not available, it is possible to interactively define the path online over the scanned surface.KeywordsRoot Mean Square ErrorControl PointRobot TrajectoryWelding PathVisibility ConstraintThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We consider the evolution of a reactive soluble substance introduced into the Poiseuille flow in a slit channel. The reactive transport happens in presence of dominant Peclet and Damkohler numbers. We suppose Peclet numbers corresponding to Taylor’s dispersion regime. The two main results of the paper are the following. First, using the anisotropic perturbation technique, we derive rigorously an effective model for the enhanced diffusion. It contains memory effects and contributions to the effective diffusion and effective advection velocity, due to the flow and chemistry reaction regime. Error estimates for the approximation of the physical solution by the upscaled one are presented in the energy norms. Presence of an initial time boundary layer allows only a global error estimate in L2 with respect to space and time. We use the Laplace’s transform in time to get optimal estimates. Second, we explicit the retardation and memory effects of the adsorption/desorption reactions on the dispersive characteristics and show their importance. The chemistry influences directly the characteristic diffusion width.

Prior work on high-order exponential operator splitting methods is extended to evolution equations defined by three linear operators. A posteriori local error estimators are constructed via a suitable integral representation of the local error involving the defect associated with the splitting solution and quadrature approximation via Hermite interpolation. In order to prove asymptotical correctness, a multiple integral representation involving iterated defects is deduced by repeated application of the variation-of-constant formula. The error analysis within the framework of abstract evolution equations provides the basis for concrete applications. Numerical examples for initial-boundary value problems of Schrödinger and of parabolic type confirm the asymptotical correctness of the proposed a posteriori error estimators.

Most methods used for the numerical solution of ordinary differential equations may be characterized by a pair of matrices $(A,B)$. In this article, local error estimates for such methods are obtained as linear combinations of computed values. (This is often equivalent to constructing pairs of embedded methods.) Although these local error estimates are obtained by assuming that exact solution values are used to start the step, it is shown that this assumption is not required for certain estimates. Some hybrid methods have such (computable) estimates, and also have simple step length change procedures.

Local and global error estimations in linear structural dynamics