Klyachin A.A., Verevkin I.Y. Construction of C1-smooth piecewise quadratic functions for solving boundary value problems of 4th order equations on a triangular grid

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https://doi.org/10.15688/mpcm.jvolsu.2023.2.1

Aleksey A. Klyachin
Doctor of Physical and Mathematical Sciences, Head of Department of Mathematical Analysis and Function Theory,
Volgograd State University
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https://orcid.org/0000-0003-3293-9066 
Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation

Ilya Y. Verevkin
Assistant Lecturer, Department of Mathematical Analysis and Function Theory,
Volgograd State University
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation

Abstract. In this paper, we present one approach to constructing continuously differentiable piecewise-quadratic functions on a triangular grid, based on smoothing a piecewise-linear function in the vicinity of edges and triangulation nodes. In the works [8], [7], the issues of approximation of the functional (1) in triangular grids and the convergence of the variational method for solving the boundary value problem of the equation (2) were studied. However, in the numerical solution there are difficulties associated with the construction of continuously differentiable piecewise polynomial functions on triangulations. In particular, their construction requires solving large systems of equations at each step of the variational method. When trying to get by with only continuous piecewise polynomial functions, we got a negative result (divergence of approximate solutions was found) [9]. In this paper, we circumvent the difficulties that arise — we indicate a method for constructing piecewise polynomial functions that have continuous partial derivatives. With the help of this class of functions, we have obtained formulas for approximating the functional (1) and tested the method on the example of a biharmonic equation.

Key words: piecewise polynomial function, variational problem, forth order equations, gradient descent method, cubic spline. 

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Construction of C1-smooth piecewise quadratic functions for solving boundary value problems of 4th order equations on a triangular grid by Klyachin A.A., Verevkin I.Y. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Mathematical Physics and Computer Simulation. Vol. 26 No. 2 2023, pp. 5-15

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