Convergence of goal-oriented adaptive finite element methods for semilinear problems
Abstract In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several related approximate dual problems that are crucial to b...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Holst, Michael [verfasserIn] Pollock, Sara [verfasserIn] Zhu, Yunrong [verfasserIn] |
|---|
| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2015 |
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| Schlagwörter: |
Adaptive finite element methods |
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| Übergeordnetes Werk: |
Enthalten in: Computing and visualization in science - Berlin : Springer, 1997, 17(2015), 1 vom: Feb., Seite 43-63 |
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| Übergeordnetes Werk: |
volume:17 ; year:2015 ; number:1 ; month:02 ; pages:43-63 |
| Links: |
|---|
| DOI / URN: |
10.1007/s00791-015-0243-1 |
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| Katalog-ID: |
SPR007851693 |
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| 245 | 1 | 0 | |a Convergence of goal-oriented adaptive finite element methods for semilinear problems |
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| 520 | |a Abstract In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several related approximate dual problems that are crucial to both the analysis as well as to the development of a practical numerical method. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We include a brief summary of a priori estimates for this class of semilinear problems, and then describe some goal-oriented variations of the standard approach to AFEM. Following the recent approach of Mommer–Stevenson and Holst–Pollock for increasingly general linear problems, we first establish a quasi-error contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined primal-dual quasi-error, and subsequently show convergence with respect to the quantity of interest. Finally, a sequence of numerical experiments are examined and it is observed that the behavior of the implementation follows the predictions of the theory. | ||
| 650 | 4 | |a Adaptive finite element methods |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Goal oriented |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Semilinear elliptic problems |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Quasi-orthogonality |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Residual-based error estimator |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Convergence |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Contraction |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a A posteriori estimates |7 (dpeaa)DE-He213 | |
| 700 | 1 | |a Pollock, Sara |e verfasserin |4 aut | |
| 700 | 1 | |a Zhu, Yunrong |e verfasserin |4 aut | |
| 773 | 0 | 8 | |i Enthalten in |t Computing and visualization in science |d Berlin : Springer, 1997 |g 17(2015), 1 vom: Feb., Seite 43-63 |w (DE-627)253722012 |w (DE-600)1458972-2 |x 1433-0369 |7 nnns |
| 773 | 1 | 8 | |g volume:17 |g year:2015 |g number:1 |g month:02 |g pages:43-63 |
| 856 | 4 | 0 | |u https://dx.doi.org/10.1007/s00791-015-0243-1 |z lizenzpflichtig |3 Volltext |
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10.1007/s00791-015-0243-1 doi (DE-627)SPR007851693 (SPR)s00791-015-0243-1-e DE-627 ger DE-627 rakwb eng 570 610 ASE 31.76 bkl 54.80 bkl 50.03 bkl 30.03 bkl Holst, Michael verfasserin aut Convergence of goal-oriented adaptive finite element methods for semilinear problems 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several related approximate dual problems that are crucial to both the analysis as well as to the development of a practical numerical method. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We include a brief summary of a priori estimates for this class of semilinear problems, and then describe some goal-oriented variations of the standard approach to AFEM. Following the recent approach of Mommer–Stevenson and Holst–Pollock for increasingly general linear problems, we first establish a quasi-error contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined primal-dual quasi-error, and subsequently show convergence with respect to the quantity of interest. Finally, a sequence of numerical experiments are examined and it is observed that the behavior of the implementation follows the predictions of the theory. Adaptive finite element methods (dpeaa)DE-He213 Goal oriented (dpeaa)DE-He213 Semilinear elliptic problems (dpeaa)DE-He213 Quasi-orthogonality (dpeaa)DE-He213 Residual-based error estimator (dpeaa)DE-He213 Convergence (dpeaa)DE-He213 Contraction (dpeaa)DE-He213 A posteriori estimates (dpeaa)DE-He213 Pollock, Sara verfasserin aut Zhu, Yunrong verfasserin aut Enthalten in Computing and visualization in science Berlin : Springer, 1997 17(2015), 1 vom: Feb., Seite 43-63 (DE-627)253722012 (DE-600)1458972-2 1433-0369 nnns volume:17 year:2015 number:1 month:02 pages:43-63 https://dx.doi.org/10.1007/s00791-015-0243-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.76 ASE 54.80 ASE 50.03 ASE 30.03 ASE AR 17 2015 1 02 43-63 |
| spelling |
10.1007/s00791-015-0243-1 doi (DE-627)SPR007851693 (SPR)s00791-015-0243-1-e DE-627 ger DE-627 rakwb eng 570 610 ASE 31.76 bkl 54.80 bkl 50.03 bkl 30.03 bkl Holst, Michael verfasserin aut Convergence of goal-oriented adaptive finite element methods for semilinear problems 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several related approximate dual problems that are crucial to both the analysis as well as to the development of a practical numerical method. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We include a brief summary of a priori estimates for this class of semilinear problems, and then describe some goal-oriented variations of the standard approach to AFEM. Following the recent approach of Mommer–Stevenson and Holst–Pollock for increasingly general linear problems, we first establish a quasi-error contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined primal-dual quasi-error, and subsequently show convergence with respect to the quantity of interest. Finally, a sequence of numerical experiments are examined and it is observed that the behavior of the implementation follows the predictions of the theory. Adaptive finite element methods (dpeaa)DE-He213 Goal oriented (dpeaa)DE-He213 Semilinear elliptic problems (dpeaa)DE-He213 Quasi-orthogonality (dpeaa)DE-He213 Residual-based error estimator (dpeaa)DE-He213 Convergence (dpeaa)DE-He213 Contraction (dpeaa)DE-He213 A posteriori estimates (dpeaa)DE-He213 Pollock, Sara verfasserin aut Zhu, Yunrong verfasserin aut Enthalten in Computing and visualization in science Berlin : Springer, 1997 17(2015), 1 vom: Feb., Seite 43-63 (DE-627)253722012 (DE-600)1458972-2 1433-0369 nnns volume:17 year:2015 number:1 month:02 pages:43-63 https://dx.doi.org/10.1007/s00791-015-0243-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.76 ASE 54.80 ASE 50.03 ASE 30.03 ASE AR 17 2015 1 02 43-63 |
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10.1007/s00791-015-0243-1 doi (DE-627)SPR007851693 (SPR)s00791-015-0243-1-e DE-627 ger DE-627 rakwb eng 570 610 ASE 31.76 bkl 54.80 bkl 50.03 bkl 30.03 bkl Holst, Michael verfasserin aut Convergence of goal-oriented adaptive finite element methods for semilinear problems 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several related approximate dual problems that are crucial to both the analysis as well as to the development of a practical numerical method. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We include a brief summary of a priori estimates for this class of semilinear problems, and then describe some goal-oriented variations of the standard approach to AFEM. Following the recent approach of Mommer–Stevenson and Holst–Pollock for increasingly general linear problems, we first establish a quasi-error contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined primal-dual quasi-error, and subsequently show convergence with respect to the quantity of interest. Finally, a sequence of numerical experiments are examined and it is observed that the behavior of the implementation follows the predictions of the theory. Adaptive finite element methods (dpeaa)DE-He213 Goal oriented (dpeaa)DE-He213 Semilinear elliptic problems (dpeaa)DE-He213 Quasi-orthogonality (dpeaa)DE-He213 Residual-based error estimator (dpeaa)DE-He213 Convergence (dpeaa)DE-He213 Contraction (dpeaa)DE-He213 A posteriori estimates (dpeaa)DE-He213 Pollock, Sara verfasserin aut Zhu, Yunrong verfasserin aut Enthalten in Computing and visualization in science Berlin : Springer, 1997 17(2015), 1 vom: Feb., Seite 43-63 (DE-627)253722012 (DE-600)1458972-2 1433-0369 nnns volume:17 year:2015 number:1 month:02 pages:43-63 https://dx.doi.org/10.1007/s00791-015-0243-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.76 ASE 54.80 ASE 50.03 ASE 30.03 ASE AR 17 2015 1 02 43-63 |
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10.1007/s00791-015-0243-1 doi (DE-627)SPR007851693 (SPR)s00791-015-0243-1-e DE-627 ger DE-627 rakwb eng 570 610 ASE 31.76 bkl 54.80 bkl 50.03 bkl 30.03 bkl Holst, Michael verfasserin aut Convergence of goal-oriented adaptive finite element methods for semilinear problems 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several related approximate dual problems that are crucial to both the analysis as well as to the development of a practical numerical method. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We include a brief summary of a priori estimates for this class of semilinear problems, and then describe some goal-oriented variations of the standard approach to AFEM. Following the recent approach of Mommer–Stevenson and Holst–Pollock for increasingly general linear problems, we first establish a quasi-error contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined primal-dual quasi-error, and subsequently show convergence with respect to the quantity of interest. Finally, a sequence of numerical experiments are examined and it is observed that the behavior of the implementation follows the predictions of the theory. Adaptive finite element methods (dpeaa)DE-He213 Goal oriented (dpeaa)DE-He213 Semilinear elliptic problems (dpeaa)DE-He213 Quasi-orthogonality (dpeaa)DE-He213 Residual-based error estimator (dpeaa)DE-He213 Convergence (dpeaa)DE-He213 Contraction (dpeaa)DE-He213 A posteriori estimates (dpeaa)DE-He213 Pollock, Sara verfasserin aut Zhu, Yunrong verfasserin aut Enthalten in Computing and visualization in science Berlin : Springer, 1997 17(2015), 1 vom: Feb., Seite 43-63 (DE-627)253722012 (DE-600)1458972-2 1433-0369 nnns volume:17 year:2015 number:1 month:02 pages:43-63 https://dx.doi.org/10.1007/s00791-015-0243-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.76 ASE 54.80 ASE 50.03 ASE 30.03 ASE AR 17 2015 1 02 43-63 |
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10.1007/s00791-015-0243-1 doi (DE-627)SPR007851693 (SPR)s00791-015-0243-1-e DE-627 ger DE-627 rakwb eng 570 610 ASE 31.76 bkl 54.80 bkl 50.03 bkl 30.03 bkl Holst, Michael verfasserin aut Convergence of goal-oriented adaptive finite element methods for semilinear problems 2015 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several related approximate dual problems that are crucial to both the analysis as well as to the development of a practical numerical method. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We include a brief summary of a priori estimates for this class of semilinear problems, and then describe some goal-oriented variations of the standard approach to AFEM. Following the recent approach of Mommer–Stevenson and Holst–Pollock for increasingly general linear problems, we first establish a quasi-error contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined primal-dual quasi-error, and subsequently show convergence with respect to the quantity of interest. Finally, a sequence of numerical experiments are examined and it is observed that the behavior of the implementation follows the predictions of the theory. Adaptive finite element methods (dpeaa)DE-He213 Goal oriented (dpeaa)DE-He213 Semilinear elliptic problems (dpeaa)DE-He213 Quasi-orthogonality (dpeaa)DE-He213 Residual-based error estimator (dpeaa)DE-He213 Convergence (dpeaa)DE-He213 Contraction (dpeaa)DE-He213 A posteriori estimates (dpeaa)DE-He213 Pollock, Sara verfasserin aut Zhu, Yunrong verfasserin aut Enthalten in Computing and visualization in science Berlin : Springer, 1997 17(2015), 1 vom: Feb., Seite 43-63 (DE-627)253722012 (DE-600)1458972-2 1433-0369 nnns volume:17 year:2015 number:1 month:02 pages:43-63 https://dx.doi.org/10.1007/s00791-015-0243-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OLC-PHA SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.76 ASE 54.80 ASE 50.03 ASE 30.03 ASE AR 17 2015 1 02 43-63 |
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Enthalten in Computing and visualization in science 17(2015), 1 vom: Feb., Seite 43-63 volume:17 year:2015 number:1 month:02 pages:43-63 |
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Holst, Michael @@aut@@ Pollock, Sara @@aut@@ Zhu, Yunrong @@aut@@ |
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<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">SPR007851693</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230520002441.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">201005s2015 xx |||||o 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s00791-015-0243-1</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR007851693</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s00791-015-0243-1-e</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="082" ind1="0" ind2="4"><subfield code="a">570</subfield><subfield code="a">610</subfield><subfield code="q">ASE</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">31.76</subfield><subfield code="2">bkl</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">54.80</subfield><subfield code="2">bkl</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">50.03</subfield><subfield code="2">bkl</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">30.03</subfield><subfield code="2">bkl</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Holst, Michael</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Convergence of goal-oriented adaptive finite element methods for semilinear problems</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2015</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several related approximate dual problems that are crucial to both the analysis as well as to the development of a practical numerical method. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We include a brief summary of a priori estimates for this class of semilinear problems, and then describe some goal-oriented variations of the standard approach to AFEM. Following the recent approach of Mommer–Stevenson and Holst–Pollock for increasingly general linear problems, we first establish a quasi-error contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined primal-dual quasi-error, and subsequently show convergence with respect to the quantity of interest. Finally, a sequence of numerical experiments are examined and it is observed that the behavior of the implementation follows the predictions of the theory.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Adaptive finite element methods</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Goal oriented</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Semilinear elliptic problems</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Quasi-orthogonality</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Residual-based error estimator</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield 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Holst, Michael |
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Holst, Michael ddc 570 bkl 31.76 bkl 54.80 bkl 50.03 bkl 30.03 misc Adaptive finite element methods misc Goal oriented misc Semilinear elliptic problems misc Quasi-orthogonality misc Residual-based error estimator misc Convergence misc Contraction misc A posteriori estimates Convergence of goal-oriented adaptive finite element methods for semilinear problems |
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570 610 ASE 31.76 bkl 54.80 bkl 50.03 bkl 30.03 bkl Convergence of goal-oriented adaptive finite element methods for semilinear problems Adaptive finite element methods (dpeaa)DE-He213 Goal oriented (dpeaa)DE-He213 Semilinear elliptic problems (dpeaa)DE-He213 Quasi-orthogonality (dpeaa)DE-He213 Residual-based error estimator (dpeaa)DE-He213 Convergence (dpeaa)DE-He213 Contraction (dpeaa)DE-He213 A posteriori estimates (dpeaa)DE-He213 |
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ddc 570 bkl 31.76 bkl 54.80 bkl 50.03 bkl 30.03 misc Adaptive finite element methods misc Goal oriented misc Semilinear elliptic problems misc Quasi-orthogonality misc Residual-based error estimator misc Convergence misc Contraction misc A posteriori estimates |
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ddc 570 bkl 31.76 bkl 54.80 bkl 50.03 bkl 30.03 misc Adaptive finite element methods misc Goal oriented misc Semilinear elliptic problems misc Quasi-orthogonality misc Residual-based error estimator misc Convergence misc Contraction misc A posteriori estimates |
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ddc 570 bkl 31.76 bkl 54.80 bkl 50.03 bkl 30.03 misc Adaptive finite element methods misc Goal oriented misc Semilinear elliptic problems misc Quasi-orthogonality misc Residual-based error estimator misc Convergence misc Contraction misc A posteriori estimates |
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Convergence of goal-oriented adaptive finite element methods for semilinear problems |
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Convergence of goal-oriented adaptive finite element methods for semilinear problems |
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Holst, Michael |
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Computing and visualization in science |
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Holst, Michael Pollock, Sara Zhu, Yunrong |
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convergence of goal-oriented adaptive finite element methods for semilinear problems |
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Convergence of goal-oriented adaptive finite element methods for semilinear problems |
| abstract |
Abstract In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several related approximate dual problems that are crucial to both the analysis as well as to the development of a practical numerical method. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We include a brief summary of a priori estimates for this class of semilinear problems, and then describe some goal-oriented variations of the standard approach to AFEM. Following the recent approach of Mommer–Stevenson and Holst–Pollock for increasingly general linear problems, we first establish a quasi-error contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined primal-dual quasi-error, and subsequently show convergence with respect to the quantity of interest. Finally, a sequence of numerical experiments are examined and it is observed that the behavior of the implementation follows the predictions of the theory. |
| abstractGer |
Abstract In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several related approximate dual problems that are crucial to both the analysis as well as to the development of a practical numerical method. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We include a brief summary of a priori estimates for this class of semilinear problems, and then describe some goal-oriented variations of the standard approach to AFEM. Following the recent approach of Mommer–Stevenson and Holst–Pollock for increasingly general linear problems, we first establish a quasi-error contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined primal-dual quasi-error, and subsequently show convergence with respect to the quantity of interest. Finally, a sequence of numerical experiments are examined and it is observed that the behavior of the implementation follows the predictions of the theory. |
| abstract_unstemmed |
Abstract In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several related approximate dual problems that are crucial to both the analysis as well as to the development of a practical numerical method. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We include a brief summary of a priori estimates for this class of semilinear problems, and then describe some goal-oriented variations of the standard approach to AFEM. Following the recent approach of Mommer–Stevenson and Holst–Pollock for increasingly general linear problems, we first establish a quasi-error contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined primal-dual quasi-error, and subsequently show convergence with respect to the quantity of interest. Finally, a sequence of numerical experiments are examined and it is observed that the behavior of the implementation follows the predictions of the theory. |
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| score |
7.399781 |