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On the convergence and rate of the convergence of a projection-difference method for approximate solving a parabolic equation with weight integral condition

Annotation

In the Hilbert space the abstract linear parabolic equation with nonlocal weight integral condition for the solution is resolved approximately by projectiondifference method using time-implicit Euler’s method. Approximation of the problem by spatial variables is oriented on the finite element method. Errors estimations of approximate solutions, convergence of approximate solution to exact one and orders of rate of convergence are established.

Keywords

Hilbert space; parabolic equation; nonlocal weighted integral condition; projection-diffrence method; time-implicit Euler’s method

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DOI

10.20310/1810-0198-2018-23-123-517-523

UDC

517.988.8

Pages

517-523

References

1. Petrova A.A., Smagin V.V. Skhodimost’ metoda Galyorkina priblizhennogo resheniya parabolicheskogo uravneniya s vesovym integral’nym usloviem na reshenie [Convergence of the Galyorkin method of approximate solving parabolic equation with weight integral condition]. Izvestiya vysshikh uchebnykh zavedeniy. Matematika – Russian Mathematics, 2016, no. 8, pp. 49-59. (In Russian). 2. Aubin J.-P. Priblizhennoe reshenie ellipticheskikh kraevykh zadach [Approximate Solution of Elliptic Boundary Problems]. Moscow, Mir Publ., 1977, 384 p. (In Russian). 3. Petrova A.A., Smagin V.V. Razreshimost’ variatsionnoy zadachi parabolicheskogo tipa s vesovym integral’nym usloviem [Solvability of the variational problem of parabolic type with a weighted integral condition]. Vestnik Voronezhskogo gosudarstvennogo universiteta. Seriya: Fizika. Matematika – Proceedings of Voronezh State University. Series: Physics. Mathematics, 2014, no. 4, pp. 160-169. (In Russian). 4. Ciarlet P.G. Metod konechnykh elementov dlya ellipticheskikh zadach [Finite Element Method for Elliptic Problems]. Moscow, Mir Publ., 1980, 512 p. (In Russian). 5. Oganesyan L.A., Rukhovets L.A. Variatsionno-raznostnye metody resheniya ellipticheskikh uravneniy [Variational-Difference Methods for Solving Elliptic Equations]. Yerevan, 1979, 236 p. (In Russian). 6. Smagin V.V. Koertsitivnye otsenki pogreshnostey proektsionnogo i proektsionno-raznostnogo metodov dlya parabolicheskikh uravneniy [The coercive estimations of errors of projection and projection-difference methods for parabolic equations]. Matematicheskiy sbornik – Sbornik: Mathematics, 1994, vol. 185, no. 11, pp. 79-94. (In Russian). 7. Lions J.-L., Magencs Е. Neodnorodnye granichnye zadachi i ikh prilozheniya [Inhomogeneous Boundary Value Problems and Their Applications]. Moscow, Mir Publ., 1971, 372 p. (In Russian).

Received

2018-04-16

Section of issue

Scientific articles

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