Schorr, Robert (2019)
Numerical Methods for Parabolic-Elliptic Interface Problems.
Technische Universität Darmstadt
Dissertation, Erstveröffentlichung
Kurzbeschreibung (Abstract)
In this thesis, we consider the numerical approximation of parabolic-elliptic interface problems with variants of the non-symmetric coupling method of MacCamy and Suri [Quart.Appl. Math., 44 (1987), pp. 675–690]. In particular, we look at the coupling of the Finite Element Method (FEM) and the Boundary Element Method (BEM) for a basic model problem and establish well-posedness and quasi-optimality of this formulation for problems with non-smooth interfaces. From this, error estimates with optimal order can be deduced. Moreover, we investigate the subsequent discretisation in time by a variant of the implicit Euler method. As for the semi-discretisation, we establish well-posedness and quasi-optimality for the fully-discrete scheme under minimal regularity assumptions on the solution. Error estimates with optimal order follow again directly.
The class of parabolic-elliptic interface problems also includes convection-dominated diffusion-convection-reaction problems. This poses a certain challenge to the solving method, as for example the Finite Element Method cannot stably solve convection-dominated problems. A possible remedy to guarantee stable solutions is the use of the vertex-centred Finite Volume Method (FVM) with an upwind stabilisation option or the Streamline Upwind Petrov Galerkin method (SUPG). The FVM has the additional advantage of the conservation of the numerical fluxes, whereas the SUPG is a simple extension of FEM. Thus, we also look at an FVM-BEM and SUPG-BEM coupling for a semi-discretisation of the underlying problem. The subsequent time-discretisation will again be achieved by the variant of the implicit Euler method. This allows us to develop an analysis under minimal regularity assumptions, not only for the semi-discrete systems but also for the fully-discrete systems.
Lastly, we show some numerical examples to illustrate our theoretical results and to give an outlook to possible practical applications, such as eddy current problems or fluid mechanics problems.
| Typ des Eintrags: | Dissertation | ||||
|---|---|---|---|---|---|
| Erschienen: | 2019 | ||||
| Autor(en): | Schorr, Robert | ||||
| Art des Eintrags: | Erstveröffentlichung | ||||
| Titel: | Numerical Methods for Parabolic-Elliptic Interface Problems | ||||
| Sprache: | Englisch | ||||
| Referenten: | Erath, Prof. Dr. Christoph ; Schäfer, Prof. Dr. Michael ; Steinbach, Prof. Dr. Olaf | ||||
| Publikationsjahr: | 2019 | ||||
| Ort: | Darmstadt | ||||
| Datum der mündlichen Prüfung: | 27 März 2019 | ||||
| URL / URN: | https://tuprints.ulb.tu-darmstadt.de/8609 | ||||
| Kurzbeschreibung (Abstract): | In this thesis, we consider the numerical approximation of parabolic-elliptic interface problems with variants of the non-symmetric coupling method of MacCamy and Suri [Quart.Appl. Math., 44 (1987), pp. 675–690]. In particular, we look at the coupling of the Finite Element Method (FEM) and the Boundary Element Method (BEM) for a basic model problem and establish well-posedness and quasi-optimality of this formulation for problems with non-smooth interfaces. From this, error estimates with optimal order can be deduced. Moreover, we investigate the subsequent discretisation in time by a variant of the implicit Euler method. As for the semi-discretisation, we establish well-posedness and quasi-optimality for the fully-discrete scheme under minimal regularity assumptions on the solution. Error estimates with optimal order follow again directly. The class of parabolic-elliptic interface problems also includes convection-dominated diffusion-convection-reaction problems. This poses a certain challenge to the solving method, as for example the Finite Element Method cannot stably solve convection-dominated problems. A possible remedy to guarantee stable solutions is the use of the vertex-centred Finite Volume Method (FVM) with an upwind stabilisation option or the Streamline Upwind Petrov Galerkin method (SUPG). The FVM has the additional advantage of the conservation of the numerical fluxes, whereas the SUPG is a simple extension of FEM. Thus, we also look at an FVM-BEM and SUPG-BEM coupling for a semi-discretisation of the underlying problem. The subsequent time-discretisation will again be achieved by the variant of the implicit Euler method. This allows us to develop an analysis under minimal regularity assumptions, not only for the semi-discrete systems but also for the fully-discrete systems. Lastly, we show some numerical examples to illustrate our theoretical results and to give an outlook to possible practical applications, such as eddy current problems or fluid mechanics problems. |
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| Alternatives oder übersetztes Abstract: |
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| URN: | urn:nbn:de:tuda-tuprints-86096 | ||||
| Sachgruppe der Dewey Dezimalklassifikatin (DDC): | 500 Naturwissenschaften und Mathematik > 510 Mathematik | ||||
| Fachbereich(e)/-gebiet(e): | Exzellenzinitiative Exzellenzinitiative > Graduiertenschulen Exzellenzinitiative > Graduiertenschulen > Graduate School of Computational Engineering (CE) 04 Fachbereich Mathematik 04 Fachbereich Mathematik > Numerik und wissenschaftliches Rechnen |
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| Hinterlegungsdatum: | 09 Jun 2019 19:55 | ||||
| Letzte Änderung: | 09 Jun 2019 19:55 | ||||
| PPN: | |||||
| Referenten: | Erath, Prof. Dr. Christoph ; Schäfer, Prof. Dr. Michael ; Steinbach, Prof. Dr. Olaf | ||||
| Datum der mündlichen Prüfung / Verteidigung / mdl. Prüfung: | 27 März 2019 | ||||
| Export: | |||||
| Suche nach Titel in: | TUfind oder in Google | ||||
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