Geodesic
On $C^1$-convergence of piecewise polynomial solutions to a fourth order variational equation
Ufa mathematical journal, Tome 14 (2022) no. 3, pp. 60-69 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In the present work we consider a boundary value problem in a polygonal domain for a fourth order variational equation. We assume that this domain is partitioned into finitely many triangles forming its triangulation. We introduce a class of piecewise polynomial functions of a given degree and for a considered equation we define the notion of a piecewise polynomial solution on a triangle net. We prove a theorem on existence and uniqueness of such solution. Moreover, we establish that under certain conditions for the triangulation of the domain, the second derivatives of the piecewise polynomial solutions are estimated by a constant independent of the fineness of the partition. This fact allows us to prove $C^1$-convergence of piecewise polynomial solutions to the equations as the fineness of grid tends to zero.
Keywords: biharmonic functions, triangular grid, piecewise polynomial function, approximation error. 
@article{UFA_2022_14_3_a6,
     author = {A. A. Klyachin},
     title = {On $C^1$-convergence of piecewise polynomial solutions to a fourth order variational equation},
     journal = {Ufa mathematical journal},
     pages = {60--69},
     year = {2022},
     volume = {14},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2022_14_3_a6/}
}
TY  - JOUR
AU  - A. A. Klyachin
TI  - On $C^1$-convergence of piecewise polynomial solutions to a fourth order variational equation
JO  - Ufa mathematical journal
PY  - 2022
SP  - 60
EP  - 69
VL  - 14
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/UFA_2022_14_3_a6/
LA  - en
ID  - UFA_2022_14_3_a6
ER  - 
%0 Journal Article
%A A. A. Klyachin
%T On $C^1$-convergence of piecewise polynomial solutions to a fourth order variational equation
%J Ufa mathematical journal
%D 2022
%P 60-69
%V 14
%N 3
%U http://geodesic.mathdoc.fr/item/UFA_2022_14_3_a6/
%G en
%F UFA_2022_14_3_a6
A. A. Klyachin. On $C^1$-convergence of piecewise polynomial solutions to a fourth order variational equation. Ufa mathematical journal, Tome 14 (2022) no. 3, pp. 60-69. http://geodesic.mathdoc.fr/item/UFA_2022_14_3_a6/

[1] Butterworth-Heinemann, Oxford, 1986 | MR

[2] P.N. Vabishchevich, “Numerical solution of variational elliptic inequalities of fourth order”, Comput. Math. Math. Phys., 24:5 (1984), 19–24 | DOI | MR | Zbl

[3] V.G. Prikazchikov, “An asymptotic estimate of the accuracy of a discrete spectral problem”, Comput. Math. Math. Phys., 31:4 (1991), 97–101 | MR | MR | Zbl

[4] G.K. Berikelashvili, “On the rate of convergence of a difference solution of the first boundary value problem for a fourth-order elliptic equation”, Differ. Equat., 35:7 (1999), 967–973 | MR | Zbl

[5] Yu.A. Bogan, “On the potential method for a fourth-order elliptic equation in anisotropic elasticity theory”, Sibir. Zhurn. Indust. Matem., 3:2 (2000), 29–34 | MR | Zbl

[6] V. V. Karachik, “Solving a problem of Robin type for biharmonic equation”, Russ. Math., 62:2 (2018), 34–48 | DOI | MR | Zbl

[7] E.A. Utkina, “Neumann problem for one equation fourth order”, Vestn. Samar. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, 2(19) (2009), 29–37 | DOI | Zbl

[8] A.L. Ushakov, “Iteration factorization for numerical solving a fourth order elliptic equation in a rectangular domain”, Vestnik Yuzhno-Ural. Univ. Ser. Matem. Mekh. Fiz., 6:1 (2014), 42–49 | Zbl

[9] M.M. Karchevsky, “Mixed method of finite elements for nonclassical boundary value problems in the theory of shallow shells”, Uchenye Zapis. Kazan. Univ. Ser. Fiz.-Matem. Nauki, 158, no. 3, 2016, 322–335

[10] V.P. Shapeev, V.A. Belyaev, “Solving the biharmonic equation with high order accuracy in irregular domains by the least squares collocation method”, Vych. Metod. Progr., 19:4 (2018), 340–355

[11] M. Ben-Artzi, I. Chorev, J.-P. Croisille, D. Fishelov, “A compact difference scheme for the biharmonic equation in planar irregular domains”, SIAM J. Numer. Anal., 47:4 (2009) | DOI | MR | Zbl

[12] I. Altas, J. Dym, M.M. Gupta, R. Manohar, “Multigrid solution of automatically generated high-order discretizations for the biharmonic equation”, SIAM J. Sci. Comput., 19 (1998), 1575–1585 | DOI | MR | Zbl

[13] A.A. Klyachin, V.A. Klyachin, “The approximation of the fourth-order partial differential equations in the class of the piecewise polynomial functions on the triangular grid”, Matem. Fiz. Kompyut. Model., 22:2 (2019), 65–72 | MR

[14] A.A. Klyachin, “Estimation of the error of calculating the functional containing higher-order derivatives on a triangular grid”, Sibir. Èlektron. Matem. Izv., 16 (2019), 1856–1867 | MR

[15] A. Ženišek, “Interpolation Polynomials on the Triangle”, Numer. Math., 15 (1970), 283–296 | DOI | MR

[16] J.H. Bramble, M. Zlamal, “Triangular elements in the finite element method”, Math. Comp., 24(112) (1970), 809–820 | DOI | MR

[17] D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Berlin, 1983 | MR | MR | Zbl