α-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation
Abstract An initial-boundary value problem of the form ${D}_{t}^{\alpha } u+{\varDelta }^{2}u-c{\varDelta } u =f$ is considered, where ${D}_{t}^{\alpha }$ is a Caputo temporal derivative of order α ∈ (0,1) and c is a nonnegative constant. The spatial domain ${\varOmega } \subset \mathbb {R}^{d}$ for...
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| Autor*in: |
Huang, Chaobao [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2020 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science+Business Media, LLC, part of Springer Nature 2020 |
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| Übergeordnetes Werk: |
Enthalten in: Numerical algorithms - Springer US, 1991, 87(2020), 4 vom: 06. Nov., Seite 1749-1766 |
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| Übergeordnetes Werk: |
volume:87 ; year:2020 ; number:4 ; day:06 ; month:11 ; pages:1749-1766 |
| Links: |
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| DOI / URN: |
10.1007/s11075-020-01036-y |
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OLC2126723011 |
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| 520 | |a Abstract An initial-boundary value problem of the form ${D}_{t}^{\alpha } u+{\varDelta }^{2}u-c{\varDelta } u =f$ is considered, where ${D}_{t}^{\alpha }$ is a Caputo temporal derivative of order α ∈ (0,1) and c is a nonnegative constant. The spatial domain ${\varOmega } \subset \mathbb {R}^{d}$ for some d ∈{1,2,3}, with Ω bounded and convex. The boundary conditions are u = Δu = 0 on ∂Ω. A priori bounds on the solution are established, given sufficient regularity and compatibility of the data; typical solutions have a weak singularity at the initial time t = 0. The problem is rewritten as a system of two second-order differential equations, then discretised using standard finite elements in space together with the L1 discretisation of ${D}_{t}^{\alpha }$ on a graded temporal mesh. The numerical method computes approximations ${u_{h}^{n}}$ and ${{p}_{h}^{n}}$ of u(⋅,tn) and Δu(⋅,tn) at each time level tn. The stability of the method (i.e. a priori bounds on $\|{{u}_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{p_{h}^{n}}\|_{L^{2}({\varOmega })}$) is established by means of a new discrete Gronwall inequality that is α-robust, i.e. remains valid as α → $ 1^{−} $. Error bounds on $\|u(\cdot , t_{n}) - {u_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{\varDelta } u(\cdot , t_{n}) - {{p}_{h}^{n}}\|_{L^{2}({\varOmega })}$ are then derived; these bounds are of optimal order in the spatial and temporal mesh parameters for each fixed value of α, and they are α-robust if one considers α → $ 1^{−} $. | ||
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10.1007/s11075-020-01036-y doi (DE-627)OLC2126723011 (DE-He213)s11075-020-01036-y-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Huang, Chaobao verfasserin (orcid)0000-0003-3554-7885 aut α-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation 2020 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2020 Abstract An initial-boundary value problem of the form ${D}_{t}^{\alpha } u+{\varDelta }^{2}u-c{\varDelta } u =f$ is considered, where ${D}_{t}^{\alpha }$ is a Caputo temporal derivative of order α ∈ (0,1) and c is a nonnegative constant. The spatial domain ${\varOmega } \subset \mathbb {R}^{d}$ for some d ∈{1,2,3}, with Ω bounded and convex. The boundary conditions are u = Δu = 0 on ∂Ω. A priori bounds on the solution are established, given sufficient regularity and compatibility of the data; typical solutions have a weak singularity at the initial time t = 0. The problem is rewritten as a system of two second-order differential equations, then discretised using standard finite elements in space together with the L1 discretisation of ${D}_{t}^{\alpha }$ on a graded temporal mesh. The numerical method computes approximations ${u_{h}^{n}}$ and ${{p}_{h}^{n}}$ of u(⋅,tn) and Δu(⋅,tn) at each time level tn. The stability of the method (i.e. a priori bounds on $\|{{u}_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{p_{h}^{n}}\|_{L^{2}({\varOmega })}$) is established by means of a new discrete Gronwall inequality that is α-robust, i.e. remains valid as α → $ 1^{−} $. Error bounds on $\|u(\cdot , t_{n}) - {u_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{\varDelta } u(\cdot , t_{n}) - {{p}_{h}^{n}}\|_{L^{2}({\varOmega })}$ are then derived; these bounds are of optimal order in the spatial and temporal mesh parameters for each fixed value of α, and they are α-robust if one considers α → $ 1^{−} $. Time-fractional problem Weak singularity Mixed finite element method Discrete Gronwall inequality -robust Stynes, Martin (orcid)0000-0003-2085-7354 aut Enthalten in Numerical algorithms Springer US, 1991 87(2020), 4 vom: 06. Nov., Seite 1749-1766 (DE-627)130981753 (DE-600)1075844-6 (DE-576)029154111 1017-1398 nnns volume:87 year:2020 number:4 day:06 month:11 pages:1749-1766 https://doi.org/10.1007/s11075-020-01036-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 87 2020 4 06 11 1749-1766 |
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10.1007/s11075-020-01036-y doi (DE-627)OLC2126723011 (DE-He213)s11075-020-01036-y-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Huang, Chaobao verfasserin (orcid)0000-0003-3554-7885 aut α-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation 2020 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2020 Abstract An initial-boundary value problem of the form ${D}_{t}^{\alpha } u+{\varDelta }^{2}u-c{\varDelta } u =f$ is considered, where ${D}_{t}^{\alpha }$ is a Caputo temporal derivative of order α ∈ (0,1) and c is a nonnegative constant. The spatial domain ${\varOmega } \subset \mathbb {R}^{d}$ for some d ∈{1,2,3}, with Ω bounded and convex. The boundary conditions are u = Δu = 0 on ∂Ω. A priori bounds on the solution are established, given sufficient regularity and compatibility of the data; typical solutions have a weak singularity at the initial time t = 0. The problem is rewritten as a system of two second-order differential equations, then discretised using standard finite elements in space together with the L1 discretisation of ${D}_{t}^{\alpha }$ on a graded temporal mesh. The numerical method computes approximations ${u_{h}^{n}}$ and ${{p}_{h}^{n}}$ of u(⋅,tn) and Δu(⋅,tn) at each time level tn. The stability of the method (i.e. a priori bounds on $\|{{u}_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{p_{h}^{n}}\|_{L^{2}({\varOmega })}$) is established by means of a new discrete Gronwall inequality that is α-robust, i.e. remains valid as α → $ 1^{−} $. Error bounds on $\|u(\cdot , t_{n}) - {u_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{\varDelta } u(\cdot , t_{n}) - {{p}_{h}^{n}}\|_{L^{2}({\varOmega })}$ are then derived; these bounds are of optimal order in the spatial and temporal mesh parameters for each fixed value of α, and they are α-robust if one considers α → $ 1^{−} $. Time-fractional problem Weak singularity Mixed finite element method Discrete Gronwall inequality -robust Stynes, Martin (orcid)0000-0003-2085-7354 aut Enthalten in Numerical algorithms Springer US, 1991 87(2020), 4 vom: 06. Nov., Seite 1749-1766 (DE-627)130981753 (DE-600)1075844-6 (DE-576)029154111 1017-1398 nnns volume:87 year:2020 number:4 day:06 month:11 pages:1749-1766 https://doi.org/10.1007/s11075-020-01036-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 87 2020 4 06 11 1749-1766 |
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10.1007/s11075-020-01036-y doi (DE-627)OLC2126723011 (DE-He213)s11075-020-01036-y-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Huang, Chaobao verfasserin (orcid)0000-0003-3554-7885 aut α-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation 2020 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2020 Abstract An initial-boundary value problem of the form ${D}_{t}^{\alpha } u+{\varDelta }^{2}u-c{\varDelta } u =f$ is considered, where ${D}_{t}^{\alpha }$ is a Caputo temporal derivative of order α ∈ (0,1) and c is a nonnegative constant. The spatial domain ${\varOmega } \subset \mathbb {R}^{d}$ for some d ∈{1,2,3}, with Ω bounded and convex. The boundary conditions are u = Δu = 0 on ∂Ω. A priori bounds on the solution are established, given sufficient regularity and compatibility of the data; typical solutions have a weak singularity at the initial time t = 0. The problem is rewritten as a system of two second-order differential equations, then discretised using standard finite elements in space together with the L1 discretisation of ${D}_{t}^{\alpha }$ on a graded temporal mesh. The numerical method computes approximations ${u_{h}^{n}}$ and ${{p}_{h}^{n}}$ of u(⋅,tn) and Δu(⋅,tn) at each time level tn. The stability of the method (i.e. a priori bounds on $\|{{u}_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{p_{h}^{n}}\|_{L^{2}({\varOmega })}$) is established by means of a new discrete Gronwall inequality that is α-robust, i.e. remains valid as α → $ 1^{−} $. Error bounds on $\|u(\cdot , t_{n}) - {u_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{\varDelta } u(\cdot , t_{n}) - {{p}_{h}^{n}}\|_{L^{2}({\varOmega })}$ are then derived; these bounds are of optimal order in the spatial and temporal mesh parameters for each fixed value of α, and they are α-robust if one considers α → $ 1^{−} $. Time-fractional problem Weak singularity Mixed finite element method Discrete Gronwall inequality -robust Stynes, Martin (orcid)0000-0003-2085-7354 aut Enthalten in Numerical algorithms Springer US, 1991 87(2020), 4 vom: 06. Nov., Seite 1749-1766 (DE-627)130981753 (DE-600)1075844-6 (DE-576)029154111 1017-1398 nnns volume:87 year:2020 number:4 day:06 month:11 pages:1749-1766 https://doi.org/10.1007/s11075-020-01036-y lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT AR 87 2020 4 06 11 1749-1766 |
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Huang, Chaobao |
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α-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation |
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α-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation |
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α-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation |
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Abstract An initial-boundary value problem of the form ${D}_{t}^{\alpha } u+{\varDelta }^{2}u-c{\varDelta } u =f$ is considered, where ${D}_{t}^{\alpha }$ is a Caputo temporal derivative of order α ∈ (0,1) and c is a nonnegative constant. The spatial domain ${\varOmega } \subset \mathbb {R}^{d}$ for some d ∈{1,2,3}, with Ω bounded and convex. The boundary conditions are u = Δu = 0 on ∂Ω. A priori bounds on the solution are established, given sufficient regularity and compatibility of the data; typical solutions have a weak singularity at the initial time t = 0. The problem is rewritten as a system of two second-order differential equations, then discretised using standard finite elements in space together with the L1 discretisation of ${D}_{t}^{\alpha }$ on a graded temporal mesh. The numerical method computes approximations ${u_{h}^{n}}$ and ${{p}_{h}^{n}}$ of u(⋅,tn) and Δu(⋅,tn) at each time level tn. The stability of the method (i.e. a priori bounds on $\|{{u}_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{p_{h}^{n}}\|_{L^{2}({\varOmega })}$) is established by means of a new discrete Gronwall inequality that is α-robust, i.e. remains valid as α → $ 1^{−} $. Error bounds on $\|u(\cdot , t_{n}) - {u_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{\varDelta } u(\cdot , t_{n}) - {{p}_{h}^{n}}\|_{L^{2}({\varOmega })}$ are then derived; these bounds are of optimal order in the spatial and temporal mesh parameters for each fixed value of α, and they are α-robust if one considers α → $ 1^{−} $. © Springer Science+Business Media, LLC, part of Springer Nature 2020 |
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Abstract An initial-boundary value problem of the form ${D}_{t}^{\alpha } u+{\varDelta }^{2}u-c{\varDelta } u =f$ is considered, where ${D}_{t}^{\alpha }$ is a Caputo temporal derivative of order α ∈ (0,1) and c is a nonnegative constant. The spatial domain ${\varOmega } \subset \mathbb {R}^{d}$ for some d ∈{1,2,3}, with Ω bounded and convex. The boundary conditions are u = Δu = 0 on ∂Ω. A priori bounds on the solution are established, given sufficient regularity and compatibility of the data; typical solutions have a weak singularity at the initial time t = 0. The problem is rewritten as a system of two second-order differential equations, then discretised using standard finite elements in space together with the L1 discretisation of ${D}_{t}^{\alpha }$ on a graded temporal mesh. The numerical method computes approximations ${u_{h}^{n}}$ and ${{p}_{h}^{n}}$ of u(⋅,tn) and Δu(⋅,tn) at each time level tn. The stability of the method (i.e. a priori bounds on $\|{{u}_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{p_{h}^{n}}\|_{L^{2}({\varOmega })}$) is established by means of a new discrete Gronwall inequality that is α-robust, i.e. remains valid as α → $ 1^{−} $. Error bounds on $\|u(\cdot , t_{n}) - {u_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{\varDelta } u(\cdot , t_{n}) - {{p}_{h}^{n}}\|_{L^{2}({\varOmega })}$ are then derived; these bounds are of optimal order in the spatial and temporal mesh parameters for each fixed value of α, and they are α-robust if one considers α → $ 1^{−} $. © Springer Science+Business Media, LLC, part of Springer Nature 2020 |
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Abstract An initial-boundary value problem of the form ${D}_{t}^{\alpha } u+{\varDelta }^{2}u-c{\varDelta } u =f$ is considered, where ${D}_{t}^{\alpha }$ is a Caputo temporal derivative of order α ∈ (0,1) and c is a nonnegative constant. The spatial domain ${\varOmega } \subset \mathbb {R}^{d}$ for some d ∈{1,2,3}, with Ω bounded and convex. The boundary conditions are u = Δu = 0 on ∂Ω. A priori bounds on the solution are established, given sufficient regularity and compatibility of the data; typical solutions have a weak singularity at the initial time t = 0. The problem is rewritten as a system of two second-order differential equations, then discretised using standard finite elements in space together with the L1 discretisation of ${D}_{t}^{\alpha }$ on a graded temporal mesh. The numerical method computes approximations ${u_{h}^{n}}$ and ${{p}_{h}^{n}}$ of u(⋅,tn) and Δu(⋅,tn) at each time level tn. The stability of the method (i.e. a priori bounds on $\|{{u}_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{p_{h}^{n}}\|_{L^{2}({\varOmega })}$) is established by means of a new discrete Gronwall inequality that is α-robust, i.e. remains valid as α → $ 1^{−} $. Error bounds on $\|u(\cdot , t_{n}) - {u_{h}^{n}}\|_{L^{2}({\varOmega })}$ and $\|{\varDelta } u(\cdot , t_{n}) - {{p}_{h}^{n}}\|_{L^{2}({\varOmega })}$ are then derived; these bounds are of optimal order in the spatial and temporal mesh parameters for each fixed value of α, and they are α-robust if one considers α → $ 1^{−} $. © Springer Science+Business Media, LLC, part of Springer Nature 2020 |
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α-robust error analysis of a mixed finite element method for a time-fractional biharmonic equation |
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