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Global Error Control For The Continuous Galerkin Finite Element Method For Ordinary Differential Equations
 M 2 AN
"... . We analyze a continuous Galerkin finite element method for the integration of initial value problems in ordinary differential equations. We derive quasioptimal a priori and a posteriori error bounds. We use these results to construct a rigorous and robust theory of global error control. We conclud ..."
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Cited by 23 (7 self)
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. We analyze a continuous Galerkin finite element method for the integration of initial value problems in ordinary differential equations. We derive quasioptimal a priori and a posteriori error bounds. We use these results to construct a rigorous and robust theory of global error control. We conclude by exhibiting the properties of the error control in a series of numerical experiments. Une m'ethode des 'el'ements finis de type Galerkin continue pour l'int'egration des probl`emes initiales pour les 'equations aux d'eriv'ees ordinaires est analys'ee. Des estimations d'erreur quasioptimales de type a priori et a posteriori sont d'emontr'ees. Les r'esultats sont employ'es dans la construction d'une th'eorie rigoureuse et robuste pour le controle globale d'erreur. La qualit'e du controle d'erreur est expos'ee dans une s'erie des exp'eriences num'eriques. x1. Introduction Our main purpose is to outline a rigorous theory of global error control for the continuous Galerkin finite element me...
AN A POSTERIORI—A PRIORI ANALYSIS OF MULTISCALE OPERATOR SPLITTING
"... Abstract. In this paper, we analyze a multiscale operator splitting method for solving systems of ordinary differential equations such as those that result upon space discretization of a reactiondiffusion equation. Our goal is to analyze and accurately estimate the error of the numerical solution, i ..."
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Abstract. In this paper, we analyze a multiscale operator splitting method for solving systems of ordinary differential equations such as those that result upon space discretization of a reactiondiffusion equation. Our goal is to analyze and accurately estimate the error of the numerical solution, including the effects of any instabilities that can result from multiscale operator splitting. We present both an a priori error analysis and a new type of hybrid a priori – a posteriori error analysis for an operator splitting discontinuous Galerkin finite element method. Both analyses clearly distinguish between the effects of the operator splitting and the discretization of each component of the decomposed problem. The hybrid analysis has the form of a computable a posteriori leading order expression and a provablyhigher order a priori expression. The hybrid analysis takes into account the fact that the adjoint problems for the original problem and a multiscale operator splitting discretization differ in significant ways. In particular, this provides the means to monitor global instabilities that can arise from operator splitting. Key words. a posteriori error analysis, adjoint problem, discontinuous Galerkin method, generalized Green’s function, goal oriented error estimates, multiscale method, operator decomposition, operator splitting, reactiondiffusion equations, residual AMS subject classifications. 65N15, 65N30, 65N50 1. Introduction. Operator
Global Dynamics of a Discontinuous Galerkin Approximation to a Class of ReactionDiffusion Equations
, 1995
"... The long time behavior of arbitrary order fully discrete approximations using the discontinuous Galerkin method (see Johnson [J]) for the time discretization of a reactiondiffusion equation is studied. The existence of absorbing sets and an attractor is shown for the numerical method. The crucial s ..."
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The long time behavior of arbitrary order fully discrete approximations using the discontinuous Galerkin method (see Johnson [J]) for the time discretization of a reactiondiffusion equation is studied. The existence of absorbing sets and an attractor is shown for the numerical method. The crucial step in the analysis involves showing the fully discrete scheme has a Lyapunov functional.
A COMPUTATIONAL MEASURE THEORETIC APPROACH TO INVERSE SENSITIVITY PROBLEMS II: A POSTERIORI ERROR ANALYSIS
"... We consider the effect of various sources of error on the inverse problem of quantifying the uncertainty of inputs to a finite dimensional map, e.g. determined implicitly by solution of a nonlinear system, given specified uncertainty in a linear functional of the output of the map. We describe the i ..."
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We consider the effect of various sources of error on the inverse problem of quantifying the uncertainty of inputs to a finite dimensional map, e.g. determined implicitly by solution of a nonlinear system, given specified uncertainty in a linear functional of the output of the map. We describe the inverse problem probabilistically, so that the uncertainty in the quantity of interest is represented by a random variable with a known distribution, and we assume that the map from the input space to the quantity of interest is smooth. We use efficient methods for determining the unique solution to the problem of inverting through a manytoone map by inverting into a quotient space representation of the input space which combines a forward sensitivity analysis with the Implicit Function Theorem, and a computational measure theoretic approach of inverting into the entire input space resulting in an approximate probability measure on the input space. There are several sources of error. There is a statistical error due to finite sampling, and two sources of deterministic errors due to approximations of the quantity of interest map. We provide an a priori result for the error resulting from linearization, and a posteriori error estimates for both the numerical error of solutions used in the linearization of the map and the error in the distribution due to sampling. Key words. a posteriori error analysis, adjoint problem, density estimation, inverse sensitivity analysis, nonparametric density estimation, sensitivity analysis 1. Introduction. In [3]