Error Analysis of a Fully Discrete Morley Finite Element Approximation for the Cahn–Hilliard Equation
Abstract This paper proposes and analyzes the Morley element method for the Cahn–Hilliard equation. It is a fourth order nonlinear singular perturbation equation arises from the binary alloy problem in materials science, and its limit is proved to approach the Hele-Shaw flow. If the %$L^2(\Omega )%$...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Li, Yukun [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2018 |
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| Schlagwörter: |
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| Übergeordnetes Werk: |
Enthalten in: Journal of scientific computing - New York, NY [u.a.] : Springer Science + Business Media B.V., 1986, 78(2018), 3 vom: 27. Sept., Seite 1862-1892 |
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| Übergeordnetes Werk: |
volume:78 ; year:2018 ; number:3 ; day:27 ; month:09 ; pages:1862-1892 |
| Links: |
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| DOI / URN: |
10.1007/s10915-018-0834-3 |
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| Katalog-ID: |
SPR014616521 |
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| 520 | |a Abstract This paper proposes and analyzes the Morley element method for the Cahn–Hilliard equation. It is a fourth order nonlinear singular perturbation equation arises from the binary alloy problem in materials science, and its limit is proved to approach the Hele-Shaw flow. If the %$L^2(\Omega )%$ error estimate is considered directly as in paper [14], we can only prove that the error bound depends on the exponential function of %$\frac{1}{\epsilon }%$. Instead, this paper derives the error bound which depends on the polynomial function of %$\frac{1}{\epsilon }%$ by considering the discrete %$H^{-1}%$ error estimate first. There are two main difficulties in proving this polynomial dependence of the discrete %$H^{-1}%$ error estimate. Firstly, it is difficult to prove discrete energy law and discrete stability results due to the complex structure of the bilinear form of the Morley element discretization. This paper overcomes this difficulty by defining four types of discrete inverse Laplace operators and exploring the relations between these discrete inverse Laplace operators and continuous inverse Laplace operator. Each of these operators plays important roles, and their relations are crucial in proving the discrete energy law, discrete stability results and error estimates. Secondly, it is difficult to prove the discrete spectrum estimate in the Morley element space because the Morley element space intersects with the %$C^1%$ conforming finite element space but they are not contained in each other. Instead of proving this discrete spectrum estimate in the Morley element space, this paper proves a generalized coercivity result by exploring properties of the enriching operators and using the discrete spectrum estimate in its %$C^1%$ conforming relative finite element space, which can be obtained by using the spectrum estimate of the Cahn–Hilliard operator. The error estimate in this paper provides an approach to prove the convergence of the numerical interfaces of the Morley element method to the interface of the Hele-Shaw flow. | ||
| 650 | 4 | |a Morley element |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Cahn–Hilliard equation |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Generalized coercivity result |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Conforming relative |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Hele-Shaw flow |7 (dpeaa)DE-He213 | |
| 773 | 0 | 8 | |i Enthalten in |t Journal of scientific computing |d New York, NY [u.a.] : Springer Science + Business Media B.V., 1986 |g 78(2018), 3 vom: 27. Sept., Seite 1862-1892 |w (DE-627)317878395 |w (DE-600)2017260-6 |x 1573-7691 |7 nnns |
| 773 | 1 | 8 | |g volume:78 |g year:2018 |g number:3 |g day:27 |g month:09 |g pages:1862-1892 |
| 856 | 4 | 0 | |u https://dx.doi.org/10.1007/s10915-018-0834-3 |z lizenzpflichtig |3 Volltext |
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10.1007/s10915-018-0834-3 doi (DE-627)SPR014616521 (SPR)s10915-018-0834-3-e DE-627 ger DE-627 rakwb eng 004 ASE 31.76 bkl 54.25 bkl Li, Yukun verfasserin aut Error Analysis of a Fully Discrete Morley Finite Element Approximation for the Cahn–Hilliard Equation 2018 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract This paper proposes and analyzes the Morley element method for the Cahn–Hilliard equation. It is a fourth order nonlinear singular perturbation equation arises from the binary alloy problem in materials science, and its limit is proved to approach the Hele-Shaw flow. If the %$L^2(\Omega )%$ error estimate is considered directly as in paper [14], we can only prove that the error bound depends on the exponential function of %$\frac{1}{\epsilon }%$. Instead, this paper derives the error bound which depends on the polynomial function of %$\frac{1}{\epsilon }%$ by considering the discrete %$H^{-1}%$ error estimate first. There are two main difficulties in proving this polynomial dependence of the discrete %$H^{-1}%$ error estimate. Firstly, it is difficult to prove discrete energy law and discrete stability results due to the complex structure of the bilinear form of the Morley element discretization. This paper overcomes this difficulty by defining four types of discrete inverse Laplace operators and exploring the relations between these discrete inverse Laplace operators and continuous inverse Laplace operator. Each of these operators plays important roles, and their relations are crucial in proving the discrete energy law, discrete stability results and error estimates. Secondly, it is difficult to prove the discrete spectrum estimate in the Morley element space because the Morley element space intersects with the %$C^1%$ conforming finite element space but they are not contained in each other. Instead of proving this discrete spectrum estimate in the Morley element space, this paper proves a generalized coercivity result by exploring properties of the enriching operators and using the discrete spectrum estimate in its %$C^1%$ conforming relative finite element space, which can be obtained by using the spectrum estimate of the Cahn–Hilliard operator. The error estimate in this paper provides an approach to prove the convergence of the numerical interfaces of the Morley element method to the interface of the Hele-Shaw flow. Morley element (dpeaa)DE-He213 Cahn–Hilliard equation (dpeaa)DE-He213 Generalized coercivity result (dpeaa)DE-He213 Conforming relative (dpeaa)DE-He213 Hele-Shaw flow (dpeaa)DE-He213 Enthalten in Journal of scientific computing New York, NY [u.a.] : Springer Science + Business Media B.V., 1986 78(2018), 3 vom: 27. Sept., Seite 1862-1892 (DE-627)317878395 (DE-600)2017260-6 1573-7691 nnns volume:78 year:2018 number:3 day:27 month:09 pages:1862-1892 https://dx.doi.org/10.1007/s10915-018-0834-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER 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_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 54.25 ASE AR 78 2018 3 27 09 1862-1892 |
| spelling |
10.1007/s10915-018-0834-3 doi (DE-627)SPR014616521 (SPR)s10915-018-0834-3-e DE-627 ger DE-627 rakwb eng 004 ASE 31.76 bkl 54.25 bkl Li, Yukun verfasserin aut Error Analysis of a Fully Discrete Morley Finite Element Approximation for the Cahn–Hilliard Equation 2018 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract This paper proposes and analyzes the Morley element method for the Cahn–Hilliard equation. It is a fourth order nonlinear singular perturbation equation arises from the binary alloy problem in materials science, and its limit is proved to approach the Hele-Shaw flow. If the %$L^2(\Omega )%$ error estimate is considered directly as in paper [14], we can only prove that the error bound depends on the exponential function of %$\frac{1}{\epsilon }%$. Instead, this paper derives the error bound which depends on the polynomial function of %$\frac{1}{\epsilon }%$ by considering the discrete %$H^{-1}%$ error estimate first. There are two main difficulties in proving this polynomial dependence of the discrete %$H^{-1}%$ error estimate. Firstly, it is difficult to prove discrete energy law and discrete stability results due to the complex structure of the bilinear form of the Morley element discretization. This paper overcomes this difficulty by defining four types of discrete inverse Laplace operators and exploring the relations between these discrete inverse Laplace operators and continuous inverse Laplace operator. Each of these operators plays important roles, and their relations are crucial in proving the discrete energy law, discrete stability results and error estimates. Secondly, it is difficult to prove the discrete spectrum estimate in the Morley element space because the Morley element space intersects with the %$C^1%$ conforming finite element space but they are not contained in each other. Instead of proving this discrete spectrum estimate in the Morley element space, this paper proves a generalized coercivity result by exploring properties of the enriching operators and using the discrete spectrum estimate in its %$C^1%$ conforming relative finite element space, which can be obtained by using the spectrum estimate of the Cahn–Hilliard operator. The error estimate in this paper provides an approach to prove the convergence of the numerical interfaces of the Morley element method to the interface of the Hele-Shaw flow. Morley element (dpeaa)DE-He213 Cahn–Hilliard equation (dpeaa)DE-He213 Generalized coercivity result (dpeaa)DE-He213 Conforming relative (dpeaa)DE-He213 Hele-Shaw flow (dpeaa)DE-He213 Enthalten in Journal of scientific computing New York, NY [u.a.] : Springer Science + Business Media B.V., 1986 78(2018), 3 vom: 27. Sept., Seite 1862-1892 (DE-627)317878395 (DE-600)2017260-6 1573-7691 nnns volume:78 year:2018 number:3 day:27 month:09 pages:1862-1892 https://dx.doi.org/10.1007/s10915-018-0834-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER 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_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 54.25 ASE AR 78 2018 3 27 09 1862-1892 |
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10.1007/s10915-018-0834-3 doi (DE-627)SPR014616521 (SPR)s10915-018-0834-3-e DE-627 ger DE-627 rakwb eng 004 ASE 31.76 bkl 54.25 bkl Li, Yukun verfasserin aut Error Analysis of a Fully Discrete Morley Finite Element Approximation for the Cahn–Hilliard Equation 2018 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract This paper proposes and analyzes the Morley element method for the Cahn–Hilliard equation. It is a fourth order nonlinear singular perturbation equation arises from the binary alloy problem in materials science, and its limit is proved to approach the Hele-Shaw flow. If the %$L^2(\Omega )%$ error estimate is considered directly as in paper [14], we can only prove that the error bound depends on the exponential function of %$\frac{1}{\epsilon }%$. Instead, this paper derives the error bound which depends on the polynomial function of %$\frac{1}{\epsilon }%$ by considering the discrete %$H^{-1}%$ error estimate first. There are two main difficulties in proving this polynomial dependence of the discrete %$H^{-1}%$ error estimate. Firstly, it is difficult to prove discrete energy law and discrete stability results due to the complex structure of the bilinear form of the Morley element discretization. This paper overcomes this difficulty by defining four types of discrete inverse Laplace operators and exploring the relations between these discrete inverse Laplace operators and continuous inverse Laplace operator. Each of these operators plays important roles, and their relations are crucial in proving the discrete energy law, discrete stability results and error estimates. Secondly, it is difficult to prove the discrete spectrum estimate in the Morley element space because the Morley element space intersects with the %$C^1%$ conforming finite element space but they are not contained in each other. Instead of proving this discrete spectrum estimate in the Morley element space, this paper proves a generalized coercivity result by exploring properties of the enriching operators and using the discrete spectrum estimate in its %$C^1%$ conforming relative finite element space, which can be obtained by using the spectrum estimate of the Cahn–Hilliard operator. The error estimate in this paper provides an approach to prove the convergence of the numerical interfaces of the Morley element method to the interface of the Hele-Shaw flow. Morley element (dpeaa)DE-He213 Cahn–Hilliard equation (dpeaa)DE-He213 Generalized coercivity result (dpeaa)DE-He213 Conforming relative (dpeaa)DE-He213 Hele-Shaw flow (dpeaa)DE-He213 Enthalten in Journal of scientific computing New York, NY [u.a.] : Springer Science + Business Media B.V., 1986 78(2018), 3 vom: 27. Sept., Seite 1862-1892 (DE-627)317878395 (DE-600)2017260-6 1573-7691 nnns volume:78 year:2018 number:3 day:27 month:09 pages:1862-1892 https://dx.doi.org/10.1007/s10915-018-0834-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER 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_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 54.25 ASE AR 78 2018 3 27 09 1862-1892 |
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10.1007/s10915-018-0834-3 doi (DE-627)SPR014616521 (SPR)s10915-018-0834-3-e DE-627 ger DE-627 rakwb eng 004 ASE 31.76 bkl 54.25 bkl Li, Yukun verfasserin aut Error Analysis of a Fully Discrete Morley Finite Element Approximation for the Cahn–Hilliard Equation 2018 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract This paper proposes and analyzes the Morley element method for the Cahn–Hilliard equation. It is a fourth order nonlinear singular perturbation equation arises from the binary alloy problem in materials science, and its limit is proved to approach the Hele-Shaw flow. If the %$L^2(\Omega )%$ error estimate is considered directly as in paper [14], we can only prove that the error bound depends on the exponential function of %$\frac{1}{\epsilon }%$. Instead, this paper derives the error bound which depends on the polynomial function of %$\frac{1}{\epsilon }%$ by considering the discrete %$H^{-1}%$ error estimate first. There are two main difficulties in proving this polynomial dependence of the discrete %$H^{-1}%$ error estimate. Firstly, it is difficult to prove discrete energy law and discrete stability results due to the complex structure of the bilinear form of the Morley element discretization. This paper overcomes this difficulty by defining four types of discrete inverse Laplace operators and exploring the relations between these discrete inverse Laplace operators and continuous inverse Laplace operator. Each of these operators plays important roles, and their relations are crucial in proving the discrete energy law, discrete stability results and error estimates. Secondly, it is difficult to prove the discrete spectrum estimate in the Morley element space because the Morley element space intersects with the %$C^1%$ conforming finite element space but they are not contained in each other. Instead of proving this discrete spectrum estimate in the Morley element space, this paper proves a generalized coercivity result by exploring properties of the enriching operators and using the discrete spectrum estimate in its %$C^1%$ conforming relative finite element space, which can be obtained by using the spectrum estimate of the Cahn–Hilliard operator. The error estimate in this paper provides an approach to prove the convergence of the numerical interfaces of the Morley element method to the interface of the Hele-Shaw flow. Morley element (dpeaa)DE-He213 Cahn–Hilliard equation (dpeaa)DE-He213 Generalized coercivity result (dpeaa)DE-He213 Conforming relative (dpeaa)DE-He213 Hele-Shaw flow (dpeaa)DE-He213 Enthalten in Journal of scientific computing New York, NY [u.a.] : Springer Science + Business Media B.V., 1986 78(2018), 3 vom: 27. Sept., Seite 1862-1892 (DE-627)317878395 (DE-600)2017260-6 1573-7691 nnns volume:78 year:2018 number:3 day:27 month:09 pages:1862-1892 https://dx.doi.org/10.1007/s10915-018-0834-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER 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_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 54.25 ASE AR 78 2018 3 27 09 1862-1892 |
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10.1007/s10915-018-0834-3 doi (DE-627)SPR014616521 (SPR)s10915-018-0834-3-e DE-627 ger DE-627 rakwb eng 004 ASE 31.76 bkl 54.25 bkl Li, Yukun verfasserin aut Error Analysis of a Fully Discrete Morley Finite Element Approximation for the Cahn–Hilliard Equation 2018 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract This paper proposes and analyzes the Morley element method for the Cahn–Hilliard equation. It is a fourth order nonlinear singular perturbation equation arises from the binary alloy problem in materials science, and its limit is proved to approach the Hele-Shaw flow. If the %$L^2(\Omega )%$ error estimate is considered directly as in paper [14], we can only prove that the error bound depends on the exponential function of %$\frac{1}{\epsilon }%$. Instead, this paper derives the error bound which depends on the polynomial function of %$\frac{1}{\epsilon }%$ by considering the discrete %$H^{-1}%$ error estimate first. There are two main difficulties in proving this polynomial dependence of the discrete %$H^{-1}%$ error estimate. Firstly, it is difficult to prove discrete energy law and discrete stability results due to the complex structure of the bilinear form of the Morley element discretization. This paper overcomes this difficulty by defining four types of discrete inverse Laplace operators and exploring the relations between these discrete inverse Laplace operators and continuous inverse Laplace operator. Each of these operators plays important roles, and their relations are crucial in proving the discrete energy law, discrete stability results and error estimates. Secondly, it is difficult to prove the discrete spectrum estimate in the Morley element space because the Morley element space intersects with the %$C^1%$ conforming finite element space but they are not contained in each other. Instead of proving this discrete spectrum estimate in the Morley element space, this paper proves a generalized coercivity result by exploring properties of the enriching operators and using the discrete spectrum estimate in its %$C^1%$ conforming relative finite element space, which can be obtained by using the spectrum estimate of the Cahn–Hilliard operator. The error estimate in this paper provides an approach to prove the convergence of the numerical interfaces of the Morley element method to the interface of the Hele-Shaw flow. Morley element (dpeaa)DE-He213 Cahn–Hilliard equation (dpeaa)DE-He213 Generalized coercivity result (dpeaa)DE-He213 Conforming relative (dpeaa)DE-He213 Hele-Shaw flow (dpeaa)DE-He213 Enthalten in Journal of scientific computing New York, NY [u.a.] : Springer Science + Business Media B.V., 1986 78(2018), 3 vom: 27. Sept., Seite 1862-1892 (DE-627)317878395 (DE-600)2017260-6 1573-7691 nnns volume:78 year:2018 number:3 day:27 month:09 pages:1862-1892 https://dx.doi.org/10.1007/s10915-018-0834-3 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER 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_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 54.25 ASE AR 78 2018 3 27 09 1862-1892 |
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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">SPR014616521</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20220111011448.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">201006s2018 xx |||||o 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s10915-018-0834-3</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR014616521</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s10915-018-0834-3-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">004</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">54.25</subfield><subfield code="2">bkl</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Li, Yukun</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Error Analysis of a Fully Discrete Morley Finite Element Approximation for the Cahn–Hilliard Equation</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2018</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 This paper proposes and analyzes the Morley element method for the Cahn–Hilliard equation. It is a fourth order nonlinear singular perturbation equation arises from the binary alloy problem in materials science, and its limit is proved to approach the Hele-Shaw flow. If the %$L^2(\Omega )%$ error estimate is considered directly as in paper [14], we can only prove that the error bound depends on the exponential function of %$\frac{1}{\epsilon }%$. Instead, this paper derives the error bound which depends on the polynomial function of %$\frac{1}{\epsilon }%$ by considering the discrete %$H^{-1}%$ error estimate first. There are two main difficulties in proving this polynomial dependence of the discrete %$H^{-1}%$ error estimate. Firstly, it is difficult to prove discrete energy law and discrete stability results due to the complex structure of the bilinear form of the Morley element discretization. This paper overcomes this difficulty by defining four types of discrete inverse Laplace operators and exploring the relations between these discrete inverse Laplace operators and continuous inverse Laplace operator. Each of these operators plays important roles, and their relations are crucial in proving the discrete energy law, discrete stability results and error estimates. Secondly, it is difficult to prove the discrete spectrum estimate in the Morley element space because the Morley element space intersects with the %$C^1%$ conforming finite element space but they are not contained in each other. Instead of proving this discrete spectrum estimate in the Morley element space, this paper proves a generalized coercivity result by exploring properties of the enriching operators and using the discrete spectrum estimate in its %$C^1%$ conforming relative finite element space, which can be obtained by using the spectrum estimate of the Cahn–Hilliard operator. The error estimate in this paper provides an approach to prove the convergence of the numerical interfaces of the Morley element method to the interface of the Hele-Shaw flow.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Morley element</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Cahn–Hilliard equation</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Generalized coercivity result</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Conforming relative</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Hele-Shaw flow</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Journal of scientific computing</subfield><subfield code="d">New York, NY [u.a.] : Springer Science + Business Media B.V., 1986</subfield><subfield code="g">78(2018), 3 vom: 27. Sept., Seite 1862-1892</subfield><subfield code="w">(DE-627)317878395</subfield><subfield code="w">(DE-600)2017260-6</subfield><subfield code="x">1573-7691</subfield><subfield code="7">nnns</subfield></datafield><datafield tag="773" ind1="1" ind2="8"><subfield code="g">volume:78</subfield><subfield code="g">year:2018</subfield><subfield code="g">number:3</subfield><subfield code="g">day:27</subfield><subfield code="g">month:09</subfield><subfield code="g">pages:1862-1892</subfield></datafield><datafield tag="856" ind1="4" ind2="0"><subfield code="u">https://dx.doi.org/10.1007/s10915-018-0834-3</subfield><subfield code="z">lizenzpflichtig</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_USEFLAG_A</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SYSFLAG_A</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_SPRINGER</subfield></datafield><datafield 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Li, Yukun |
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Li, Yukun ddc 004 bkl 31.76 bkl 54.25 misc Morley element misc Cahn–Hilliard equation misc Generalized coercivity result misc Conforming relative misc Hele-Shaw flow Error Analysis of a Fully Discrete Morley Finite Element Approximation for the Cahn–Hilliard Equation |
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004 ASE 31.76 bkl 54.25 bkl Error Analysis of a Fully Discrete Morley Finite Element Approximation for the Cahn–Hilliard Equation Morley element (dpeaa)DE-He213 Cahn–Hilliard equation (dpeaa)DE-He213 Generalized coercivity result (dpeaa)DE-He213 Conforming relative (dpeaa)DE-He213 Hele-Shaw flow (dpeaa)DE-He213 |
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Error Analysis of a Fully Discrete Morley Finite Element Approximation for the Cahn–Hilliard Equation |
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Error Analysis of a Fully Discrete Morley Finite Element Approximation for the Cahn–Hilliard Equation |
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error analysis of a fully discrete morley finite element approximation for the cahn–hilliard equation |
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Error Analysis of a Fully Discrete Morley Finite Element Approximation for the Cahn–Hilliard Equation |
| abstract |
Abstract This paper proposes and analyzes the Morley element method for the Cahn–Hilliard equation. It is a fourth order nonlinear singular perturbation equation arises from the binary alloy problem in materials science, and its limit is proved to approach the Hele-Shaw flow. If the %$L^2(\Omega )%$ error estimate is considered directly as in paper [14], we can only prove that the error bound depends on the exponential function of %$\frac{1}{\epsilon }%$. Instead, this paper derives the error bound which depends on the polynomial function of %$\frac{1}{\epsilon }%$ by considering the discrete %$H^{-1}%$ error estimate first. There are two main difficulties in proving this polynomial dependence of the discrete %$H^{-1}%$ error estimate. Firstly, it is difficult to prove discrete energy law and discrete stability results due to the complex structure of the bilinear form of the Morley element discretization. This paper overcomes this difficulty by defining four types of discrete inverse Laplace operators and exploring the relations between these discrete inverse Laplace operators and continuous inverse Laplace operator. Each of these operators plays important roles, and their relations are crucial in proving the discrete energy law, discrete stability results and error estimates. Secondly, it is difficult to prove the discrete spectrum estimate in the Morley element space because the Morley element space intersects with the %$C^1%$ conforming finite element space but they are not contained in each other. Instead of proving this discrete spectrum estimate in the Morley element space, this paper proves a generalized coercivity result by exploring properties of the enriching operators and using the discrete spectrum estimate in its %$C^1%$ conforming relative finite element space, which can be obtained by using the spectrum estimate of the Cahn–Hilliard operator. The error estimate in this paper provides an approach to prove the convergence of the numerical interfaces of the Morley element method to the interface of the Hele-Shaw flow. |
| abstractGer |
Abstract This paper proposes and analyzes the Morley element method for the Cahn–Hilliard equation. It is a fourth order nonlinear singular perturbation equation arises from the binary alloy problem in materials science, and its limit is proved to approach the Hele-Shaw flow. If the %$L^2(\Omega )%$ error estimate is considered directly as in paper [14], we can only prove that the error bound depends on the exponential function of %$\frac{1}{\epsilon }%$. Instead, this paper derives the error bound which depends on the polynomial function of %$\frac{1}{\epsilon }%$ by considering the discrete %$H^{-1}%$ error estimate first. There are two main difficulties in proving this polynomial dependence of the discrete %$H^{-1}%$ error estimate. Firstly, it is difficult to prove discrete energy law and discrete stability results due to the complex structure of the bilinear form of the Morley element discretization. This paper overcomes this difficulty by defining four types of discrete inverse Laplace operators and exploring the relations between these discrete inverse Laplace operators and continuous inverse Laplace operator. Each of these operators plays important roles, and their relations are crucial in proving the discrete energy law, discrete stability results and error estimates. Secondly, it is difficult to prove the discrete spectrum estimate in the Morley element space because the Morley element space intersects with the %$C^1%$ conforming finite element space but they are not contained in each other. Instead of proving this discrete spectrum estimate in the Morley element space, this paper proves a generalized coercivity result by exploring properties of the enriching operators and using the discrete spectrum estimate in its %$C^1%$ conforming relative finite element space, which can be obtained by using the spectrum estimate of the Cahn–Hilliard operator. The error estimate in this paper provides an approach to prove the convergence of the numerical interfaces of the Morley element method to the interface of the Hele-Shaw flow. |
| abstract_unstemmed |
Abstract This paper proposes and analyzes the Morley element method for the Cahn–Hilliard equation. It is a fourth order nonlinear singular perturbation equation arises from the binary alloy problem in materials science, and its limit is proved to approach the Hele-Shaw flow. If the %$L^2(\Omega )%$ error estimate is considered directly as in paper [14], we can only prove that the error bound depends on the exponential function of %$\frac{1}{\epsilon }%$. Instead, this paper derives the error bound which depends on the polynomial function of %$\frac{1}{\epsilon }%$ by considering the discrete %$H^{-1}%$ error estimate first. There are two main difficulties in proving this polynomial dependence of the discrete %$H^{-1}%$ error estimate. Firstly, it is difficult to prove discrete energy law and discrete stability results due to the complex structure of the bilinear form of the Morley element discretization. This paper overcomes this difficulty by defining four types of discrete inverse Laplace operators and exploring the relations between these discrete inverse Laplace operators and continuous inverse Laplace operator. Each of these operators plays important roles, and their relations are crucial in proving the discrete energy law, discrete stability results and error estimates. Secondly, it is difficult to prove the discrete spectrum estimate in the Morley element space because the Morley element space intersects with the %$C^1%$ conforming finite element space but they are not contained in each other. Instead of proving this discrete spectrum estimate in the Morley element space, this paper proves a generalized coercivity result by exploring properties of the enriching operators and using the discrete spectrum estimate in its %$C^1%$ conforming relative finite element space, which can be obtained by using the spectrum estimate of the Cahn–Hilliard operator. The error estimate in this paper provides an approach to prove the convergence of the numerical interfaces of the Morley element method to the interface of the Hele-Shaw flow. |
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| title_short |
Error Analysis of a Fully Discrete Morley Finite Element Approximation for the Cahn–Hilliard Equation |
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https://dx.doi.org/10.1007/s10915-018-0834-3 |
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| doi_str |
10.1007/s10915-018-0834-3 |
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2024-07-04T02:27:21.308Z |
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| score |
7.402128 |