<?xml version="1.0" encoding="utf-8"?>
<export-example>
  <doc>
    <id>597</id>
    <completedYear>2025</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>1</belongsToBibliography>
    <completedDate>2025-02-12</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">A posteriori error control for a finite volume scheme for a cross-diffusion model of ion transport</title>
    <abstract language="eng">We derive a reliable a posteriori error estimate for a cell-centered finite volume scheme approximating a cross-diffusion system modeling ion transport through nanopores. To this end we derive an abstract stability framework that is independent of the numerical scheme and introduce a suitable (conforming) reconstruction of the numerical solution. The stability framework relies on some simplifying assumption that coincide with those made in weak uniqueness results for this system. This is the first a posteriori error estimate for a cross-diffusion system. Along the way, we derive a pointwise a posteriori error estimate for a finite volume scheme approximating the diffusion equation. We conduct numerical experiments showing that the error estimator scales with the same order as the true error.</abstract>
    <enrichment key="SubmissionStatus">under review</enrichment>
    <enrichment key="opus.source">publish</enrichment>
    <licence>Creative Commons - CC BY-NC-SA - Namensnennung - Nicht kommerziell -  Weitergabe unter gleichen Bedingungen 4.0 International</licence>
    <author>Arne Berrens</author>
    <author>Jan Giesselmann</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>cross-diffusion</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>ion transport</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>finite-volume approximation</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>a posteriori error estimates</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>diffusion equation</value>
    </subject>
    <collection role="institutes" number="">Technische Universität Darmstadt</collection>
    <collection role="subprojects" number="">C05</collection>
    <file>https://opus4.kobv.de/opus4-trr154/files/597/BerrensGiesselmann2025.pdf</file>
  </doc>
  <doc>
    <id>539</id>
    <completedYear>2023</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>1</belongsToBibliography>
    <completedDate>2024-02-13</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Theory of shifts, shocks, and the intimate connections to L2-type a posteriori error analysis of numerical schemes for hyperbolic problems</title>
    <abstract language="eng">In this paper, we develop reliable a posteriori error estimates for numerical approximations of scalar hyperbolic conservation laws in one space dimension.&#13;
Our methods have no inherent small-data limitations and are a step towards error control of numerical schemes for systems. We are careful not to appeal to the Kruzhkov theory for scalar conservation laws. Instead, we derive novel quantitative stability estimates that extend the theory of shifts, and in particular, the framework for proving stability first developed by the second author and Vasseur. This is the first time this methodology has been used for quantitative estimates.&#13;
We work entirely within the context of the theory of shifts and a-contraction, techniques which adapt well to systems. In fact, the stability framework by the second author and Vasseur has itself recently been pushed to systems [Chen-Krupa-Vasseur. Uniqueness and weak-BV stability for 2×2 conservation laws. Arch. Ration. Mech. Anal., 246(1):299--332, 2022].&#13;
Our theoretical findings are complemented by a numerical implementation in MATLAB and numerical experiments.</abstract>
    <enrichment key="SubmissionStatus">under review</enrichment>
    <enrichment key="opus.source">publish</enrichment>
    <licence>Creative Commons - CC BY-NC-ND - Namensnennung - Nicht kommerziell - Keine Bearbeitungen 4.0 International</licence>
    <author>Jan Giesselmann</author>
    <author>Sam Krupa</author>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Conservation laws</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>entropy conditions</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>entropy solutions</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>shocks,</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>a posteriori error estimates</value>
    </subject>
    <collection role="institutes" number="">Technische Universität Darmstadt</collection>
    <collection role="subprojects" number="">C05</collection>
    <file>https://opus4.kobv.de/opus4-trr154/files/539/4-shifts-shocks.pdf</file>
  </doc>
</export-example>
