A posteriori error analysis of round-off errors in the numerical solution of ordinary differential equations

We prove sharp, computable error estimates for the propagation of errors in the numerical solution of ordinary differential equations. The new estimates extend previous estimates of the influence of data errors and discretization errors with a new term accounting for the propagation of numerical round-off errors, showing that the accumulated round-off error is inversely proportional to the square root of the step size. As a consequence, the numeric precision eventually sets the limit for the pointwise computability of accurate solutions of any ODE. The theoretical results are supported by numerically computed solutions and error estimates for the Lorenz system and the van der Pol oscillator.     

Error propagation of general linear methods for ordinary differential equations

We discuss error propagation for general linear methods for ordinary differential equations up to terms of order p+2, where p is the order of the method. These results are then applied to the estimation of local discretization errors for methods of order p and for the adjacent order p+1. The results of numerical experiments confirm the reliability of these estimates. This research has applications in the design of robust stepsize and order changing strategies for algorithms based on general linear methods.     

Method of lines with boundary elements for 1‐D transient diffusion‐reaction problems

Time‐dependent differential equations can be solved using the concept of method of lines (MOL) together with the boundary element (BE) representation for the spatial linear part of the equation. The BE method alleviates the need for spatial discretization and casts the problem in an integral format. Hence errors associated with the numerical approximation of the spatial derivatives are totally eliminated. An element level local cubic approximation is used for the variable at each time step to facilitate the time marching and the nonlinear terms are represented in a semi‐implicit manner by a local linearization at each time step. The accuracy of the method has been illustrated on a number of test problems of engineering significance. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

An adaptive edge element method and its convergence for a Saddle‐Point problem from magnetostatics

Reliable and efficient a posteriori error estimates are derived for the edge element discretization of a saddle‐point Maxwell's system. By means of the error estimates, an adaptive edge element method is proposed and its convergence is rigorously demonstrated. The algorithm uses a marking strategy based only on the error indicators, without the commonly used information on local oscillations and the refinement to meet the standard interior node property. Some new ingredients in the analysis include a novel quasi‐orthogonality and a new inf‐sup inequality associated with an appropriately chosen norm. It is shown that the algorithm is a contraction for the sum of the energy error plus the error indicators after each refinement step. Numerical experiments are presented to show the robustness and effectiveness of the proposed adaptive algorithm. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

A Reliable Error Estimation for Diagonally Implicit Multistage Integration Methods

A new approach to the estimation of the local discretization error for diagonally implicit multistage integration methods (DIMSIMs) is described. The error estimates that are obtained are very accurate and very reliable for both explicit and implicit methods for any stepsize pattern.This revised version was published online in October 2005 with corrections to the Cover Date.     

Error Estimation for Discontinuous Galerkin Solutions of Two-Dimensional Hyperbolic Problems

We analyze the discretization errors of discontinuous Galerkin solutions of steady two-dimensional hyperbolic conservation laws on unstructured meshes. We show that the leading term of the error on each element is a linear combination of orthogonal polynomials of degrees p and p+1. We further show that there is a strong superconvergence property at the outflow edge(s) of each element where the average discretization error converges as O(h ^2p+1) compared to a global rate of O(h ^p+1). Our analyses apply to both linear and nonlinear conservation laws with smooth solutions. We show how to use our theory to construct efficient and asymptotically exact a posteriori discretization error estimates and we apply these to some examples.     

High-order adaptive finite element-singly implicit Runge-Kutta methods for parabolic differential equations

We describe an adaptive mesh refinement finite element method-of-lines procedure for solving one-dimensional parabolic partial differential equations. Solutions are calculated using Galerkin's method with a piecewise hierarchical polynomial basis in space and singly implicit Runge-Kutta (SIRK) methods in time. A modified SIRK formulation eliminates a linear systems solution that is required by the traditional SIRK formulation and leads to a new reduced-order interpolation formula. Stability and temporal error estimation techniques allow acceptance of approximate solutions at intermediate stages, yielding increased efficiency when solving partial differential equations. A priori energy estimates of the local discretization error are obtained for a nonlinear scalar problem. A posteriori estimates of local spatial discretization errors, obtained by order variation, are used with the a priori error estimates to control the adaptive mesh refinement strategy. Computational results suggest convergence of the a posteriori error estimate to the exact discretization error and verify the utility of the adaptive technique.This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR-90-0194; the U.S. Army Research Office under Contract Number DAAL 03-91-G-0215; by the National Science Foundation under Grant Number CDA-8805910; and by a grant from the Committee on Research, Tulane University.     

Unconditionally Stable General Linear Methods for Ordinary Differential Equations

We describe a new approach to the construction of general linear methods with inherent Runge–Kutta stability properties, which are unconditionally stable for any stepsize pattern. These methods permit an accurate, efficient and reliable estimation of the local discretization errors. This is confirmed by numerical experiments.     

Analysis of stability and error estimates for three methods approximating a nonlinear reaction-diffusion equation

We present the error analysis of three time-stepping schemes used in the discretization of a nonlinear reaction-diffusion equation with Neumann boundary conditions, relevant in phase transition. We prove $L^\infty$ stability by maximum principle arguments, and derive error estimates using energy methods for the implicit Euler, and two implicit-explicit approaches, a linearized scheme and a fractional step method. A numerical experiment validates the theoretical results, comparing the accuracy of the methods.     

Explicit Nordsieck methods with extended stability regions

We describe the construction of explicit Nordsieck methods of order p and stage order q = p with large regions of absolute stability. We also discuss error propagation and estimation of local discretization errors. The error estimators are derived for examples of general linear methods constructed in this paper. Some numerical experiments are presented which illustrate the effectiveness of proposed methods.     

A control reduced primal interior point method for a class of control constrained optimal control problems

A primal interior point method for control constrained optimal control problems with PDE constraints is considered. Pointwise elimination of the control leads to a homotopy in the remaining state and dual variables, which is addressed by a short step pathfollowing method. The algorithm is applied to the continuous, infinite dimensional problem, where discretization is performed only in the innermost loop when solving linear equations. The a priori elimination of the least regular control permits to obtain the required accuracy with comparatively coarse meshes. Convergence of the method and discretization errors are studied, and the method is illustrated at two numerical examples. Supported by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin. This paper appeared as ZIB Report 04-38.     

A local discontinuous Galerkin method for pattern formation dynamical model in polymerizing action flocks

In this paper, we apply local discontinuous Galerkin methods to the pattern formation dynamical model in polymerizing action flocks. Optimal error estimates for the density and filament polarization in different norms are established. We use a semi-implicit spectral deferred correction time method for time discretization, which allows a relative large time step and avoids computation of a Jacobian matrix. Numerical experiments are presented to verify the theoretical analysis and to show the capability for simulations of action wave formation.

     

Localized pointwise error estimates for mixed finite element methods

In this paper we give weighted, or localized, pointwise error estimates which are valid for two different mixed finite element methods for a general second-order linear elliptic problem and for general choices of mixed elements for simplicial meshes. These estimates, similar in spirit to those recently proved by Schatz for the basic Galerkin finite element method for elliptic problems, show that the dependence of the pointwise errors in both the scalar and vector variables on the derivative of the solution is mostly local in character or conversely that the global dependence of the pointwise errors is weak. This localization is more pronounced for higher order elements. Our estimates indicate that localization occurs except when the lowest order Brezzi-Douglas-Marini elements are used, and we provide computational examples showing that the error is indeed not localized when these elements are employed.

     

Local error estimates for moderately smooth problems: Part II—SDEs and SDAEs with small noise

The paper consists of two parts. In the first part of the paper, we proposed 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 developed local error estimates for the case when the problem data is only moderately smooth, which is typically the case in stochastic differential equations. In this second part, we will consider the estimation of local errors in context of mean-square convergent methods for stochastic differential equations (SDEs) with small noise and index-1 stochastic differential-algebraic equations (SDAEs). Numerical experiments illustrate the performance of the mesh adaptation based on the local error estimation developed in this paper. The first author acknowledges support by the BMBF-project 03RONAVN and the second author support by the Austrian Science Fund Project P17253.     

Error analysis of fully discrete velocity-correction methods for incompressible flows

A fully discrete version of the velocity-correction method, proposed by Guermond and Shen (2003) for the time-dependent Navier-Stokes equations, is introduced and analyzed. It is shown that, when accounting for space discretization, additional consistency terms, which vanish when space is not discretized, have to be added to establish stability and optimal convergence. Error estimates are derived for both the standard version and the rotational version of the method. These error estimates are consistent with those by Guermond and Shen (2003) as far as time discretiztion is concerned and are optimal in space for finite elements satisfying the inf-sup condition.

     

Power bounded prolongations and Picard-Lindelöf iteration

Summary The possibility of balancing the iteration and discretization errors in iterative solution of large systems of initial value problems is discussed. The main result answers the question affirmatively by stating that in the convergence process, in a model situation, the decay rate of iteration errors is essentially independent of the step size refinement process.     

On the Convergence Analysis of the Inexact Linearly Implicit Euler Scheme for a Class of Stochastic Partial Differential Equations

This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe’s method. We use the implicit Euler scheme for the time discretization. Consequently, in each step, an elliptic equation with random right-hand side has to be solved. In practice, this cannot be performed exactly, so that efficient numerical methods are needed. Well-established adaptive wavelet or finite-element schemes, which are guaranteed to converge with optimal order, suggest themselves. We investigate how the errors corresponding to the adaptive spatial discretization propagate in time, and we show how in each time step the tolerances have to be chosen such that the resulting perturbed discretization scheme realizes the same order of convergence as the one with exact evaluations of the elliptic subproblems.     

Solving parabolic problems using local defect correction in combination with the finite volume method

We present a method for solving partial differential equations characterized by highly localized properties in which the local defect correction (LDC) algorithm for time‐dependent problems is combined with a finite volume discretization. At each time step, LDC computes a numerical solution on a composite grid, a union of a global uniform coarse grid and a local uniform fine grid. The main feature of the method is that the discrete conservation property, typical of the finite volume approach is preserved on the composite grid. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Space/time convergence analysis of a class of conservative schemes for linear wave equations

This paper concerns the space/time convergence analysis of conservative two-step time discretizations for linear wave equations. Explicit and implicit, second- and fourth-order schemes are considered, while the space discretization is given and satisfies minimal hypotheses. Convergence analysis is done using energy techniques and holds if the time step is upper-bounded by a quantity depending on space discretization parameters. In addition to showing the convergence for recently introduced fourth-order schemes, the novelty of this work consists in the independency of the convergence estimates with respect to the difference between the time step and its greatest admissible value.     

Uniform and high-order discretization schemes for Sturm–Liouville problems via Fer streamers

The current paper concerns the uniform and high-order discretization of the novel approach to the computation of Sturm–Liouville problems via Fer streamers, put forth in Ramos and Iserles (Numer. Math. 131(3), 541—565 2015). In particular, the discretization schemes are shown to enjoy large step sizes uniform over the entire eigenvalue range and tight error estimates uniform for every eigenvalue. They are made explicit for global orders 4,7,10. In addition, the present paper provides total error estimates that quantify the interplay between the truncation and the discretization in the approach by Fer streamers.