title: Efficient goal-oriented global error  estimation for BDF-type methods using  discrete adjoints
creator: Beigel, Dörte
subject: 510
subject: 510 Mathematics
description: This thesis develops estimation techniques for the global error that occurs during the approximation of solutions of Initial Value Problems (IVPs) on given intervals by multistep integration methods based on Backward Differentiation Formulas (BDF). To this end, discrete adjoints obtained by adjoint Internal Numerical  Differentiation (IND) of the nominal integration scheme are used. For this purpose, a bridge between BDF methods and Petrov-Galerkin Finite Element (FE) methods is built by a novel functional-analytic framework. Goal-oriented global error estimators are derived in analogy to the Dual Weighted Residual methodology in Galerkin methods for Partial Differential Equations. Their asymptotic behavior, their accuracy in BDF methods with variable order and stepsize as well as their applicability for global error control are investigated.  The novel results presented in this thesis include:   i) a functional-analytic framework for IVPs in Ordinary Differential Equations (ODEs) in the Banach space of continuously differentiable functions. This framework is needed since the classical Hilbert space setting is not suitable to analyze the relation between the discrete values of the adjoint IND scheme and the solution of the adjoint IVP. The new framework gives rise to the definition  of weak solutions of adjoint IVPs.   ii) a Petrov-Galerkin FE discretization of the function spaces that allows to transform the variational formulations of the IVP and of its adjoint IVP into finite  dimensional problems. The equivalence of these finite dimensional problems to BDF methods with variable but prescribed order and stepsize and their adjoint IND schemes is shown. Thus, the FE approximation of the weak adjoint is  determined by the discrete values of the adjoint IND scheme and discretization and differentiation commute in the developed framework.  iii) a proof that the values of the adjoint IND scheme corresponding to a BDF method with constant order and stepsize converge to the solution of the adjoint IVP on the open interval. In addition, a proof is given that demonstrates the convergence of the FE approximation to the weak solution of the adjoint IVP on the entire interval.  iv) goal-oriented global error estimators for BDF methods that weight, for each integration step, a local error quantity with the corresponding value of the adjoint IND scheme and yields in sum an accurate and efficient estimate for the actual error. As local error quantity defect integrals and local truncation errors are employed, respectively.  v) strategies for goal-oriented global error control in BDF methods that either adapt the locally acting relative tolerance or the given integration scheme using the stepwise error indicators.  vi) an ODE model of an exothermic, self-accelerating chemical reaction with mass transfer carried out in a discontinuous Stirred Tank Reactor. With this real-world example from chemical engineering the applicability and reliability of the novel techniques for the approximation of weak adjoints and for the simulation with goal-oriented global error control are shown.
date: 2012
type: Dissertation
type: info:eu-repo/semantics/doctoralThesis
type: NonPeerReviewed
format: application/pdf
identifier: https://archiv.ub.uni-heidelberg.de/volltextserverhttps://archiv.ub.uni-heidelberg.de/volltextserver/14317/1/Dissertation_Beigel.pdf
identifier: DOI:10.11588/heidok.00014317
identifier: urn:nbn:de:bsz:16-heidok-143177
identifier:   Beigel, Dörte  (2012) Efficient goal-oriented global error estimation for BDF-type methods using discrete adjoints.  [Dissertation]     
relation: https://archiv.ub.uni-heidelberg.de/volltextserver/14317/
rights: info:eu-repo/semantics/openAccess
rights: http://archiv.ub.uni-heidelberg.de/volltextserver/help/license_urhg.html
language: eng