A direct discontinuous Galerkin method for a time-fractional diffusion equation with a Robin boundary condition
A time-fractional reaction–diffusion initial-boundary value problem with Robin boundary condition is considered on the domain Ω × [ 0 , T ] , where...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Huang, Chaobao [verfasserIn] Stynes, Martin [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: Applied numerical mathematics - Amsterdam [u.a.] : Elsevier, 1985, 135, Seite 15-29 |
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| Übergeordnetes Werk: |
volume:135 ; pages:15-29 |
| DOI / URN: |
10.1016/j.apnum.2018.08.006 |
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| Katalog-ID: |
ELV00082190X |
|---|
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| 024 | 7 | |a 10.1016/j.apnum.2018.08.006 |2 doi | |
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| 100 | 1 | |a Huang, Chaobao |e verfasserin |0 (orcid)0000-0003-3554-7885 |4 aut | |
| 245 | 1 | 0 | |a A direct discontinuous Galerkin method for a time-fractional diffusion equation with a Robin boundary condition |
| 264 | 1 | |c 2018 | |
| 336 | |a nicht spezifiziert |b zzz |2 rdacontent | ||
| 337 | |a Computermedien |b c |2 rdamedia | ||
| 338 | |a Online-Ressource |b cr |2 rdacarrier | ||
| 520 | |a A time-fractional reaction–diffusion initial-boundary value problem with Robin boundary condition is considered on the domain Ω × [ 0 , T ] , where Ω = ( 0 , l ) ⊂ R . The coefficient of the zero-order reaction term is not required to be non-negative, which complicates the analysis. In general the unknown solution will have a weak singularity at the initial time t = 0 . Existence and uniqueness of the solution and pointwise bounds on some of its derivatives are derived. A fully discrete numerical method for computing an approximate solution is investigated; it uses the well-known L1 discretisation on a graded mesh in time and a direct discontinuous Galerkin (DDG) finite element method on a uniform mesh in space. Discrete stability of the computed solution is proved. Its error is bounded in the L 2 ( Ω ) and H 1 ( Ω ) norms at each discrete time level t n by means of a non-trivial projection of the unknown solution into the finite element space. The L 2 ( Ω ) bound is optimal for all t n ; the H 1 ( Ω ) bound is optimal for t n not close to t = 0 . An optimal grading of the temporal mesh can be deduced from these bounds. Numerical results show that our analysis is sharp. | ||
| 650 | 4 | |a DDG method | |
| 650 | 4 | |a Fractional reaction–diffusion equation | |
| 650 | 4 | |a Robin boundary condition | |
| 650 | 4 | |a Graded mesh | |
| 700 | 1 | |a Stynes, Martin |e verfasserin |0 (orcid)0000-0003-2085-7354 |4 aut | |
| 773 | 0 | 8 | |i Enthalten in |t Applied numerical mathematics |d Amsterdam [u.a.] : Elsevier, 1985 |g 135, Seite 15-29 |h Online-Ressource |w (DE-627)266888879 |w (DE-600)1468770-7 |w (DE-576)075962314 |7 nnns |
| 773 | 1 | 8 | |g volume:135 |g pages:15-29 |
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| 936 | b | k | |a 31.76 |j Numerische Mathematik |
| 951 | |a AR | ||
| 952 | |d 135 |h 15-29 | ||
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10.1016/j.apnum.2018.08.006 doi (DE-627)ELV00082190X (ELSEVIER)S0168-9274(18)30174-0 DE-627 ger DE-627 rda eng 510 DE-600 31.76 bkl Huang, Chaobao verfasserin (orcid)0000-0003-3554-7885 aut A direct discontinuous Galerkin method for a time-fractional diffusion equation with a Robin boundary condition 2018 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier A time-fractional reaction–diffusion initial-boundary value problem with Robin boundary condition is considered on the domain Ω × [ 0 , T ] , where Ω = ( 0 , l ) ⊂ R . The coefficient of the zero-order reaction term is not required to be non-negative, which complicates the analysis. In general the unknown solution will have a weak singularity at the initial time t = 0 . Existence and uniqueness of the solution and pointwise bounds on some of its derivatives are derived. A fully discrete numerical method for computing an approximate solution is investigated; it uses the well-known L1 discretisation on a graded mesh in time and a direct discontinuous Galerkin (DDG) finite element method on a uniform mesh in space. Discrete stability of the computed solution is proved. Its error is bounded in the L 2 ( Ω ) and H 1 ( Ω ) norms at each discrete time level t n by means of a non-trivial projection of the unknown solution into the finite element space. The L 2 ( Ω ) bound is optimal for all t n ; the H 1 ( Ω ) bound is optimal for t n not close to t = 0 . An optimal grading of the temporal mesh can be deduced from these bounds. Numerical results show that our analysis is sharp. DDG method Fractional reaction–diffusion equation Robin boundary condition Graded mesh Stynes, Martin verfasserin (orcid)0000-0003-2085-7354 aut Enthalten in Applied numerical mathematics Amsterdam [u.a.] : Elsevier, 1985 135, Seite 15-29 Online-Ressource (DE-627)266888879 (DE-600)1468770-7 (DE-576)075962314 nnns volume:135 pages:15-29 GBV_USEFLAG_U SYSFLAG_U GBV_ELV SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 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_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4313 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4393 31.76 Numerische Mathematik AR 135 15-29 |
| spelling |
10.1016/j.apnum.2018.08.006 doi (DE-627)ELV00082190X (ELSEVIER)S0168-9274(18)30174-0 DE-627 ger DE-627 rda eng 510 DE-600 31.76 bkl Huang, Chaobao verfasserin (orcid)0000-0003-3554-7885 aut A direct discontinuous Galerkin method for a time-fractional diffusion equation with a Robin boundary condition 2018 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier A time-fractional reaction–diffusion initial-boundary value problem with Robin boundary condition is considered on the domain Ω × [ 0 , T ] , where Ω = ( 0 , l ) ⊂ R . The coefficient of the zero-order reaction term is not required to be non-negative, which complicates the analysis. In general the unknown solution will have a weak singularity at the initial time t = 0 . Existence and uniqueness of the solution and pointwise bounds on some of its derivatives are derived. A fully discrete numerical method for computing an approximate solution is investigated; it uses the well-known L1 discretisation on a graded mesh in time and a direct discontinuous Galerkin (DDG) finite element method on a uniform mesh in space. Discrete stability of the computed solution is proved. Its error is bounded in the L 2 ( Ω ) and H 1 ( Ω ) norms at each discrete time level t n by means of a non-trivial projection of the unknown solution into the finite element space. The L 2 ( Ω ) bound is optimal for all t n ; the H 1 ( Ω ) bound is optimal for t n not close to t = 0 . An optimal grading of the temporal mesh can be deduced from these bounds. Numerical results show that our analysis is sharp. DDG method Fractional reaction–diffusion equation Robin boundary condition Graded mesh Stynes, Martin verfasserin (orcid)0000-0003-2085-7354 aut Enthalten in Applied numerical mathematics Amsterdam [u.a.] : Elsevier, 1985 135, Seite 15-29 Online-Ressource (DE-627)266888879 (DE-600)1468770-7 (DE-576)075962314 nnns volume:135 pages:15-29 GBV_USEFLAG_U SYSFLAG_U GBV_ELV SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 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_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4313 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4393 31.76 Numerische Mathematik AR 135 15-29 |
| allfields_unstemmed |
10.1016/j.apnum.2018.08.006 doi (DE-627)ELV00082190X (ELSEVIER)S0168-9274(18)30174-0 DE-627 ger DE-627 rda eng 510 DE-600 31.76 bkl Huang, Chaobao verfasserin (orcid)0000-0003-3554-7885 aut A direct discontinuous Galerkin method for a time-fractional diffusion equation with a Robin boundary condition 2018 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier A time-fractional reaction–diffusion initial-boundary value problem with Robin boundary condition is considered on the domain Ω × [ 0 , T ] , where Ω = ( 0 , l ) ⊂ R . The coefficient of the zero-order reaction term is not required to be non-negative, which complicates the analysis. In general the unknown solution will have a weak singularity at the initial time t = 0 . Existence and uniqueness of the solution and pointwise bounds on some of its derivatives are derived. A fully discrete numerical method for computing an approximate solution is investigated; it uses the well-known L1 discretisation on a graded mesh in time and a direct discontinuous Galerkin (DDG) finite element method on a uniform mesh in space. Discrete stability of the computed solution is proved. Its error is bounded in the L 2 ( Ω ) and H 1 ( Ω ) norms at each discrete time level t n by means of a non-trivial projection of the unknown solution into the finite element space. The L 2 ( Ω ) bound is optimal for all t n ; the H 1 ( Ω ) bound is optimal for t n not close to t = 0 . An optimal grading of the temporal mesh can be deduced from these bounds. Numerical results show that our analysis is sharp. DDG method Fractional reaction–diffusion equation Robin boundary condition Graded mesh Stynes, Martin verfasserin (orcid)0000-0003-2085-7354 aut Enthalten in Applied numerical mathematics Amsterdam [u.a.] : Elsevier, 1985 135, Seite 15-29 Online-Ressource (DE-627)266888879 (DE-600)1468770-7 (DE-576)075962314 nnns volume:135 pages:15-29 GBV_USEFLAG_U SYSFLAG_U GBV_ELV SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 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_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4313 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4393 31.76 Numerische Mathematik AR 135 15-29 |
| allfieldsGer |
10.1016/j.apnum.2018.08.006 doi (DE-627)ELV00082190X (ELSEVIER)S0168-9274(18)30174-0 DE-627 ger DE-627 rda eng 510 DE-600 31.76 bkl Huang, Chaobao verfasserin (orcid)0000-0003-3554-7885 aut A direct discontinuous Galerkin method for a time-fractional diffusion equation with a Robin boundary condition 2018 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier A time-fractional reaction–diffusion initial-boundary value problem with Robin boundary condition is considered on the domain Ω × [ 0 , T ] , where Ω = ( 0 , l ) ⊂ R . The coefficient of the zero-order reaction term is not required to be non-negative, which complicates the analysis. In general the unknown solution will have a weak singularity at the initial time t = 0 . Existence and uniqueness of the solution and pointwise bounds on some of its derivatives are derived. A fully discrete numerical method for computing an approximate solution is investigated; it uses the well-known L1 discretisation on a graded mesh in time and a direct discontinuous Galerkin (DDG) finite element method on a uniform mesh in space. Discrete stability of the computed solution is proved. Its error is bounded in the L 2 ( Ω ) and H 1 ( Ω ) norms at each discrete time level t n by means of a non-trivial projection of the unknown solution into the finite element space. The L 2 ( Ω ) bound is optimal for all t n ; the H 1 ( Ω ) bound is optimal for t n not close to t = 0 . An optimal grading of the temporal mesh can be deduced from these bounds. Numerical results show that our analysis is sharp. DDG method Fractional reaction–diffusion equation Robin boundary condition Graded mesh Stynes, Martin verfasserin (orcid)0000-0003-2085-7354 aut Enthalten in Applied numerical mathematics Amsterdam [u.a.] : Elsevier, 1985 135, Seite 15-29 Online-Ressource (DE-627)266888879 (DE-600)1468770-7 (DE-576)075962314 nnns volume:135 pages:15-29 GBV_USEFLAG_U SYSFLAG_U GBV_ELV SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 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_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4313 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4393 31.76 Numerische Mathematik AR 135 15-29 |
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a direct discontinuous galerkin method for a time-fractional diffusion equation with a robin boundary condition |
| title_auth |
A direct discontinuous Galerkin method for a time-fractional diffusion equation with a Robin boundary condition |
| abstract |
A time-fractional reaction–diffusion initial-boundary value problem with Robin boundary condition is considered on the domain Ω × [ 0 , T ] , where Ω = ( 0 , l ) ⊂ R . The coefficient of the zero-order reaction term is not required to be non-negative, which complicates the analysis. In general the unknown solution will have a weak singularity at the initial time t = 0 . Existence and uniqueness of the solution and pointwise bounds on some of its derivatives are derived. A fully discrete numerical method for computing an approximate solution is investigated; it uses the well-known L1 discretisation on a graded mesh in time and a direct discontinuous Galerkin (DDG) finite element method on a uniform mesh in space. Discrete stability of the computed solution is proved. Its error is bounded in the L 2 ( Ω ) and H 1 ( Ω ) norms at each discrete time level t n by means of a non-trivial projection of the unknown solution into the finite element space. The L 2 ( Ω ) bound is optimal for all t n ; the H 1 ( Ω ) bound is optimal for t n not close to t = 0 . An optimal grading of the temporal mesh can be deduced from these bounds. Numerical results show that our analysis is sharp. |
| abstractGer |
A time-fractional reaction–diffusion initial-boundary value problem with Robin boundary condition is considered on the domain Ω × [ 0 , T ] , where Ω = ( 0 , l ) ⊂ R . The coefficient of the zero-order reaction term is not required to be non-negative, which complicates the analysis. In general the unknown solution will have a weak singularity at the initial time t = 0 . Existence and uniqueness of the solution and pointwise bounds on some of its derivatives are derived. A fully discrete numerical method for computing an approximate solution is investigated; it uses the well-known L1 discretisation on a graded mesh in time and a direct discontinuous Galerkin (DDG) finite element method on a uniform mesh in space. Discrete stability of the computed solution is proved. Its error is bounded in the L 2 ( Ω ) and H 1 ( Ω ) norms at each discrete time level t n by means of a non-trivial projection of the unknown solution into the finite element space. The L 2 ( Ω ) bound is optimal for all t n ; the H 1 ( Ω ) bound is optimal for t n not close to t = 0 . An optimal grading of the temporal mesh can be deduced from these bounds. Numerical results show that our analysis is sharp. |
| abstract_unstemmed |
A time-fractional reaction–diffusion initial-boundary value problem with Robin boundary condition is considered on the domain Ω × [ 0 , T ] , where Ω = ( 0 , l ) ⊂ R . The coefficient of the zero-order reaction term is not required to be non-negative, which complicates the analysis. In general the unknown solution will have a weak singularity at the initial time t = 0 . Existence and uniqueness of the solution and pointwise bounds on some of its derivatives are derived. A fully discrete numerical method for computing an approximate solution is investigated; it uses the well-known L1 discretisation on a graded mesh in time and a direct discontinuous Galerkin (DDG) finite element method on a uniform mesh in space. Discrete stability of the computed solution is proved. Its error is bounded in the L 2 ( Ω ) and H 1 ( Ω ) norms at each discrete time level t n by means of a non-trivial projection of the unknown solution into the finite element space. The L 2 ( Ω ) bound is optimal for all t n ; the H 1 ( Ω ) bound is optimal for t n not close to t = 0 . An optimal grading of the temporal mesh can be deduced from these bounds. Numerical results show that our analysis is sharp. |
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| title_short |
A direct discontinuous Galerkin method for a time-fractional diffusion equation with a Robin boundary condition |
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