|
On $C^1$-convergence of piecewise polynomial solutions to a fourth order variational equation
A. A. Klyachin Volgograd State University
Abstract:
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.
Received: 18.05.2022
Citation:
A. A. Klyachin, “On $C^1$-convergence of piecewise polynomial solutions to a fourth order variational equation”, Ufa Math. J., 14:3 (2022), 60–69
Linking options:
https://www.mathnet.ru/eng/ufa622https://doi.org/10.13108/2022-14-3-60 https://www.mathnet.ru/eng/ufa/v14/i3/p63
|
Statistics & downloads: |
Abstract page: | 78 | Russian version PDF: | 22 | English version PDF: | 16 | References: | 18 |
|