An adaptive multilevel approach to parabolic equations in two space dimensions:
Abstract: "A new adaptive multilevel approach for linear parabolic partial differential equations is presented, which is able to handle complicated space geometries, discontinuous coefficients, inconsistent initial data. Discretization in time first (Rothe's method) with order and stepsize...
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | German |
| Veröffentlicht: |
Berlin
1991
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| Schriftenreihe: | Konrad-Zuse-Zentrum für Informationstechnik <Berlin>: Technical report
1991,7 |
| Schlagworte: | |
| Zusammenfassung: | Abstract: "A new adaptive multilevel approach for linear parabolic partial differential equations is presented, which is able to handle complicated space geometries, discontinuous coefficients, inconsistent initial data. Discretization in time first (Rothe's method) with order and stepsize control is perturbed by an adaptive finite element discretization of the elliptic subproblems, whose errors are controlled independently. Thus the high standards of solving adaptively ordinary differential equations and elliptic boundary value problems are combined. A theory of time discretization in Hilbert space is developed which yields to an optimal variable order method based on a multiplicative error correction The problem of an efficient solution of the singularly perturbed elliptic subproblems and the problem of error estimation for them can be uniquely solved within the framework of preconditioning. A multilevel nodal basis preconditioner is derived, which allows the use of highly nonuniform triangulations. Implementation issues are discussed in detail. Numerous numerical examples in one and two space dimensions clearly show the significant perspectives opened by the new algorithmic approach. Finally an application of the method is given in the area of hyperthermia, a recent clinical method for cancer therapy. |
| Beschreibung: | Zugl.: Berlin, Univ., Diss., 1991 |
| Beschreibung: | II, 136 S. graph. Darst. |
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| 490 | 1 | |a Konrad-Zuse-Zentrum für Informationstechnik <Berlin>: Technical report |v 1991,7 | |
| 500 | |a Zugl.: Berlin, Univ., Diss., 1991 | ||
| 520 | 3 | |a Abstract: "A new adaptive multilevel approach for linear parabolic partial differential equations is presented, which is able to handle complicated space geometries, discontinuous coefficients, inconsistent initial data. Discretization in time first (Rothe's method) with order and stepsize control is perturbed by an adaptive finite element discretization of the elliptic subproblems, whose errors are controlled independently. Thus the high standards of solving adaptively ordinary differential equations and elliptic boundary value problems are combined. A theory of time discretization in Hilbert space is developed which yields to an optimal variable order method based on a multiplicative error correction | |
| 520 | 3 | |a The problem of an efficient solution of the singularly perturbed elliptic subproblems and the problem of error estimation for them can be uniquely solved within the framework of preconditioning. A multilevel nodal basis preconditioner is derived, which allows the use of highly nonuniform triangulations. Implementation issues are discussed in detail. Numerous numerical examples in one and two space dimensions clearly show the significant perspectives opened by the new algorithmic approach. Finally an application of the method is given in the area of hyperthermia, a recent clinical method for cancer therapy. | |
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Datensatz im Suchindex
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| author | Bornemann, Folkmar 1967- |
| author_GND | (DE-588)120096269 |
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| author_sort | Bornemann, Folkmar 1967- |
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| bvnumber | BV004581757 |
| classification_tum | MAT 672d MAT 674d |
| ctrlnum | (OCoLC)25422130 (DE-599)BVBBV004581757 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV004581757 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T16:14:35Z |
| institution | BVB |
| language | German |
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| physical | II, 136 S. graph. Darst. |
| publishDate | 1991 |
| publishDateSearch | 1991 |
| publishDateSort | 1991 |
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| series | Konrad-Zuse-Zentrum für Informationstechnik <Berlin>: Technical report |
| series2 | Konrad-Zuse-Zentrum für Informationstechnik <Berlin>: Technical report |
| spelling | Bornemann, Folkmar 1967- Verfasser (DE-588)120096269 aut An adaptive multilevel approach to parabolic equations in two space dimensions Folkmar A. Bornemann Berlin 1991 II, 136 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Konrad-Zuse-Zentrum für Informationstechnik <Berlin>: Technical report 1991,7 Zugl.: Berlin, Univ., Diss., 1991 Abstract: "A new adaptive multilevel approach for linear parabolic partial differential equations is presented, which is able to handle complicated space geometries, discontinuous coefficients, inconsistent initial data. Discretization in time first (Rothe's method) with order and stepsize control is perturbed by an adaptive finite element discretization of the elliptic subproblems, whose errors are controlled independently. Thus the high standards of solving adaptively ordinary differential equations and elliptic boundary value problems are combined. A theory of time discretization in Hilbert space is developed which yields to an optimal variable order method based on a multiplicative error correction The problem of an efficient solution of the singularly perturbed elliptic subproblems and the problem of error estimation for them can be uniquely solved within the framework of preconditioning. A multilevel nodal basis preconditioner is derived, which allows the use of highly nonuniform triangulations. Implementation issues are discussed in detail. Numerous numerical examples in one and two space dimensions clearly show the significant perspectives opened by the new algorithmic approach. Finally an application of the method is given in the area of hyperthermia, a recent clinical method for cancer therapy. Differential equations, Parabolic Finite-Elemente-Methode (DE-588)4017233-8 gnd rswk-swf Lineare parabolische Differentialgleichung (DE-588)4221396-4 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Lineare parabolische Differentialgleichung (DE-588)4221396-4 s DE-604 Finite-Elemente-Methode (DE-588)4017233-8 s Konrad-Zuse-Zentrum für Informationstechnik <Berlin>: Technical report 1991,7 (DE-604)BV005567559 1991,7 |
| spellingShingle | Bornemann, Folkmar 1967- An adaptive multilevel approach to parabolic equations in two space dimensions Konrad-Zuse-Zentrum für Informationstechnik <Berlin>: Technical report Differential equations, Parabolic Finite-Elemente-Methode (DE-588)4017233-8 gnd Lineare parabolische Differentialgleichung (DE-588)4221396-4 gnd |
| subject_GND | (DE-588)4017233-8 (DE-588)4221396-4 (DE-588)4113937-9 |
| title | An adaptive multilevel approach to parabolic equations in two space dimensions |
| title_auth | An adaptive multilevel approach to parabolic equations in two space dimensions |
| title_exact_search | An adaptive multilevel approach to parabolic equations in two space dimensions |
| title_full | An adaptive multilevel approach to parabolic equations in two space dimensions Folkmar A. Bornemann |
| title_fullStr | An adaptive multilevel approach to parabolic equations in two space dimensions Folkmar A. Bornemann |
| title_full_unstemmed | An adaptive multilevel approach to parabolic equations in two space dimensions Folkmar A. Bornemann |
| title_short | An adaptive multilevel approach to parabolic equations in two space dimensions |
| title_sort | an adaptive multilevel approach to parabolic equations in two space dimensions |
| topic | Differential equations, Parabolic Finite-Elemente-Methode (DE-588)4017233-8 gnd Lineare parabolische Differentialgleichung (DE-588)4221396-4 gnd |
| topic_facet | Differential equations, Parabolic Finite-Elemente-Methode Lineare parabolische Differentialgleichung Hochschulschrift |
| volume_link | (DE-604)BV005567559 |
| work_keys_str_mv | AT bornemannfolkmar anadaptivemultilevelapproachtoparabolicequationsintwospacedimensions |