Convergence of the Projection-Difference Method for the Approximate Solution of a Smoothly Solvable Parabolic Equation with a Weighted Integral Condition
We search for an approximate solution of an abstract linear parabolic equation in a Hilbert space with a nonlocal weighted integral condition by the projection-difference method and the implicit Euler method in time. The approximation of the problem with respect to spatial variables is oriented to t...
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| Published in | Journal of mathematical sciences (New York, N.Y.) Vol. 275; no. 5; pp. 583 - 591 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.10.2023
Springer Springer Nature B.V |
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| ISSN | 1072-3374 1573-8795 1573-8795 |
| DOI | 10.1007/s10958-023-06699-1 |
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| Abstract | We search for an approximate solution of an abstract linear parabolic equation in a Hilbert space with a nonlocal weighted integral condition by the projection-difference method and the implicit Euler method in time. The approximation of the problem with respect to spatial variables is oriented to the finite element method in the case of arbitrary projection subspaces under an additional smoothness condition. Estimates of errors of approximate solutions are established, the convergence of approximate solutions to the exact solution is proved, and the convergence rate is estimated. |
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| AbstractList | We search for an approximate solution of an abstract linear parabolic equation in a Hilbert space with a nonlocal weighted integral condition by the projection-difference method and the implicit Euler method in time. The approximation of the problem with respect to spatial variables is oriented to the finite element method in the case of arbitrary projection subspaces under an additional smoothness condition. Estimates of errors of approximate solutions are established, the convergence of approximate solutions to the exact solution is proved, and the convergence rate is estimated. We search for an approximate solution of an abstract linear parabolic equation in a Hilbert space with a nonlocal weighted integral condition by the projection-difference method and the implicit Euler method in time. The approximation of the problem with respect to spatial variables is oriented to the finite element method in the case of arbitrary projection subspaces under an additional smoothness condition. Estimates of errors of approximate solutions are established, the convergence of approximate solutions to the exact solution is proved, and the convergence rate is estimated. Keywords and phrases: Hilbert space, parabolic equation, nonlocal weighted integral condition, projection-difference method, implicit Euler method. AMS Subject Classification: 35K90 |
| Audience | Academic |
| Author | Petrova, A. A. |
| Author_xml | – sequence: 1 givenname: A. A. surname: Petrova fullname: Petrova, A. A. email: rezolwenta@mail.ru organization: Voronezh State University |
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| Cites_doi | 10.1115/1.3424474 10.4213/sm214 10.1134/S0374064118070142 |
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| Keywords | nonlocal weighted integral condition Hilbert space parabolic equation implicit Euler method 35K90 projection-difference method |
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| References | LionsJ-LMagenesEProblèmes aux limites non homogènes et applications1971ParisDunod0212.43801 P. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam–New York–Oxford (1978). PetrovaAAConvergence of the projection-difference method of an approximate solution of a parabolic equation with a weighted integral condition on the solutionDiffer. Uravn.20185479759873849333 SmaginVVEstimates of the convergence rate of the projection and projection-difference methods for weakly solvable parabolic equationsMat. Sb.199718831431601462027 MarchukGIAgoshkovVIIntroduction to Projection-Grid Methods1981MoscowNauka0642.65037[in Russian] A. A. Petrova and V. V. Smagin, “Solvability of a parabolic variational problem type with a weighted integral condition,” Vestn. Voronezh. Univ. Ser. Fiz. Mat., No. 4, 160–169 (2014). SamarskiiAALazarovRDMakarovVLFinite-Difference Schemes for Differential Equations with Generalized Solutions1987MoscowVysshaya Shkola[in Russian] V. V. Smagin, “Energy convergence of the error of the projection-difference method for weakly solvable parabolic equations,” Tr. Mat. Fak. Voronezh. Univ., No. 4, 114–119 (1999). SmaginVVProjection-difference methods for approximate solution of parabolic equations with nonsymmetric operatorsDiffer. Uravn.200137111512318464060996.65084 AubinJPApproximation of Elliptic Boundary-Value Problems1972New York-LondonWiley-Interscience0248.65063 VainikkoGMOiaPEOn the convergence and the rate of convergence of the Galerkin method for abstract evolutionary equationsDiffer. Uravn.197511712691277 JP Aubin (6699_CR1) 1972 GI Marchuk (6699_CR4) 1981 AA Samarskii (6699_CR7) 1987 6699_CR2 6699_CR9 6699_CR6 AA Petrova (6699_CR5) 2018; 54 VV Smagin (6699_CR8) 1997; 188 GM Vainikko (6699_CR11) 1975; 11 J-L Lions (6699_CR3) 1971 VV Smagin (6699_CR10) 2001; 37 |
| References_xml | – reference: SamarskiiAALazarovRDMakarovVLFinite-Difference Schemes for Differential Equations with Generalized Solutions1987MoscowVysshaya Shkola[in Russian] – reference: MarchukGIAgoshkovVIIntroduction to Projection-Grid Methods1981MoscowNauka0642.65037[in Russian] – reference: A. A. Petrova and V. V. Smagin, “Solvability of a parabolic variational problem type with a weighted integral condition,” Vestn. Voronezh. Univ. Ser. Fiz. Mat., No. 4, 160–169 (2014). – reference: V. V. Smagin, “Energy convergence of the error of the projection-difference method for weakly solvable parabolic equations,” Tr. Mat. Fak. Voronezh. Univ., No. 4, 114–119 (1999). – reference: SmaginVVProjection-difference methods for approximate solution of parabolic equations with nonsymmetric operatorsDiffer. Uravn.200137111512318464060996.65084 – reference: AubinJPApproximation of Elliptic Boundary-Value Problems1972New York-LondonWiley-Interscience0248.65063 – reference: P. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam–New York–Oxford (1978). – reference: VainikkoGMOiaPEOn the convergence and the rate of convergence of the Galerkin method for abstract evolutionary equationsDiffer. Uravn.197511712691277 – reference: SmaginVVEstimates of the convergence rate of the projection and projection-difference methods for weakly solvable parabolic equationsMat. Sb.199718831431601462027 – reference: LionsJ-LMagenesEProblèmes aux limites non homogènes et applications1971ParisDunod0212.43801 – reference: PetrovaAAConvergence of the projection-difference method of an approximate solution of a parabolic equation with a weighted integral condition on the solutionDiffer. Uravn.20185479759873849333 – volume-title: Approximation of Elliptic Boundary-Value Problems year: 1972 ident: 6699_CR1 – ident: 6699_CR6 – ident: 6699_CR2 doi: 10.1115/1.3424474 – volume: 11 start-page: 1269 issue: 7 year: 1975 ident: 6699_CR11 publication-title: Differ. Uravn. – volume: 188 start-page: 143 issue: 3 year: 1997 ident: 6699_CR8 publication-title: Mat. Sb. doi: 10.4213/sm214 – volume-title: Problèmes aux limites non homogènes et applications year: 1971 ident: 6699_CR3 – volume: 54 start-page: 975 issue: 7 year: 2018 ident: 6699_CR5 publication-title: Differ. Uravn. doi: 10.1134/S0374064118070142 – ident: 6699_CR9 – volume-title: Introduction to Projection-Grid Methods year: 1981 ident: 6699_CR4 – volume-title: Finite-Difference Schemes for Differential Equations with Generalized Solutions year: 1987 ident: 6699_CR7 – volume: 37 start-page: 115 issue: 1 year: 2001 ident: 6699_CR10 publication-title: Differ. Uravn. |
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| SubjectTerms | Convergence Exact solutions Finite element method Hilbert space Mathematics Mathematics and Statistics Smoothness Subspaces |
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| Title | Convergence of the Projection-Difference Method for the Approximate Solution of a Smoothly Solvable Parabolic Equation with a Weighted Integral Condition |
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