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An Adaptive Algorithm for Ordinary, Stochastic and Partial Differential Equations
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.ORCID iD: 0000-0002-2964-7225
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
2005 (English)In: Recent Advances in Adaptive Computation, Providence: American Mathematical Society , 2005, 325-343 p.Chapter in book (Other academic)
Abstract [en]

The theory of a posteriori error estimates suitable for adaptive refinement is well established. This work focuses on the fundamental, but less studied, issue of convergence rates of adaptive algorithms. In particular, this work describes a simple and general adaptive algorithm applied to ordinary, stochastic and partial differential equations with proven convergence rates. The presentation has three parts: The error approximations used to build error indicators for the adaptive algorithm are based on error expansions with computable leading order terms. It is explained how to measure optimal convergence rates for approximation of functionals of the solution, and why convergence of the error density is always useful and subtle in the case of stochastic and partial differential equations. The adaptive algorithm, performing successive mesh refinements, either reduces the maximal error indicator by a factor or stops with the error asymptotically bounded by the prescribed accuracy requirement. Furthermore, the algorithm stops using the optimal number of degrees of freedom, up to a problem independent factor.

Place, publisher, year, edition, pages
Providence: American Mathematical Society , 2005. 325-343 p.
Series
Contemporary mathematics, ISSN 0271-4132 ; 383
Keyword [en]
FINITE-ELEMENT METHODS; CONVERGENCE-RATES; ERROR; APPROXIMATION; ESTIMATORS
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-7420ISBN: 0-8218-3662-5 (print)OAI: oai:DiVA.org:kth-7420DiVA: diva2:12443
Note
QC 20100824Available from: 2007-08-31 Created: 2007-08-31 Last updated: 2010-08-24Bibliographically approved
In thesis
1. Adaptivity for Stochastic and Partial Differential Equations with Applications to Phase Transformations
Open this publication in new window or tab >>Adaptivity for Stochastic and Partial Differential Equations with Applications to Phase Transformations
2007 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

his work is concentrated on efforts to efficiently compute properties of systems, modelled by differential equations, involving multiple scales. Goal oriented adaptivity is the common approach to all the treated problems. Here the goal of a numerical computation is to approximate a functional of the solution to the differential equation and the numerical method is adapted to this task.

The thesis consists of four papers. The first three papers concern the convergence of adaptive algorithms for numerical solution of differential equations; based on a posteriori expansions of global errors in the sought functional, the discretisations used in a numerical solution of the differential equiation are adaptively refined. The fourth paper uses expansion of the adaptive modelling error to compute a stochastic differential equation for a phase-field by coarse-graining molecular dynamics.

An adaptive algorithm aims to minimise the number of degrees of freedom to make the error in the functional less than a given tolerance. The number of degrees of freedom provides the convergence rate of the adaptive algorithm as the tolerance tends to zero. Provided that the computational work is proportional to the degrees of freedom this gives an estimate of the efficiency of the algorithm.

The first paper treats approximation of functionals of solutions to second order elliptic partial differential equations in bounded domains of ℝd, using isoparametric $d$-linear quadrilateral finite elements. For an adaptive algorithm, an error expansion with computable leading order term is derived %. and used in a computable error density, which is proved to converge uniformly as the mesh size tends to zero. For each element an error indicator is defined by the computed error density multiplying the local mesh size to the power of 2+d. The adaptive algorithm is based on successive subdivisions of elements, where it uses the error indicators. It is proved, using the uniform convergence of the error density, that the algorithm either reduces the maximal error indicator with a factor or stops; if it stops, then the error is asymptotically bounded by the tolerance using the optimal number of elements for an adaptive isotropic mesh, up to a problem independent factor. Here the optimal number of elements is proportional to the d/2 power of the Ldd+2

quasi-norm of the error density, whereas a uniform mesh requires a number of elements proportional to the d/2 power of the larger L1 norm of the same error density to obtain the same accuracy. For problems with multiple scales, in particular, these convergence rates may differ much, even though the convergence order may be the same.

The second paper presents an adaptive algorithm for Monte Carlo Euler approximation of the expected value E[g(X(τ),\τ)] of a given function g depending on the solution X of an \Ito\ stochastic differential equation and on the first exit time τ from a given domain. An error expansion with computable leading order term for the approximation of E[g(X(T))] with a fixed final time T>0 was given in~[Szepessy, Tempone, and Zouraris, Comm. Pure and Appl. Math., 54, 1169-1214, 2001]. This error expansion is now extended to the case with stopped diffusion. In the extension conditional probabilities are used to estimate the first exit time error, and difference quotients are used to approximate the initial data of the dual solutions. For the stopped diffusion problem the time discretisation error is of order N-1/2 for a method with N uniform time steps. Numerical results show that the adaptive algorithm improves the time discretisation error to the order N-1, with N adaptive time steps.

The third paper gives an overview of the application of the adaptive algorithm in the first two papers to ordinary, stochastic, and partial differential equation. The fourth paper investigates the possibility of computing some of the model functions in an Allen--Cahn type phase-field equation from a microscale model, where the material is described by stochastic, Smoluchowski, molecular dynamics. A local average of contributions to the potential energy in the micro model is used to determine the local phase, and a stochastic phase-field model is computed by coarse-graining the molecular dynamics. Molecular dynamics simulations on a two phase system at the melting point are used to compute a double-well reaction term in the Allen--Cahn equation and a diffusion matrix describing the noise in the coarse-grained phase-field.

Place, publisher, year, edition, pages
Stockholm: KTH, 2007. x, 26 p.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2007:12
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-4477 (URN)978-91-7178-744-6 (ISBN)
Public defence
2007-09-17, F3, Lindstedtsvägen 26, Stockholm, 13:00
Opponent
Supervisors
Note
QC 20100823Available from: 2007-08-31 Created: 2007-08-31 Last updated: 2010-09-23Bibliographically approved

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von Schwerin, Erik

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