
Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2016, Issue 5(36), Pages 124–139
(Mi vvgum136)




Mathematics
On the convergence of almost polynomial solutions of the minimal surface
I. V. Truhlyaeva^{} ^{} Volgograd State University
Abstract:
In this paper we consider the polynomial approximate solutions of the Dirichlet
problem for minimal surface equation. It is shown that under certain conditions on the
geometric structure of the domain the absolute values of the gradients of the solutions are
bounded as the degree of these polynomials increases. The obtained properties imply the
uniform convergence of approximate solutions to the exact solution of the minimal surface
equation.
In numerical solving of boundary value problems for equations and systems of partial differential equations, a very important issue is the convergence of approximate solutions. The study
of this issue is especially important for nonlinear equations since in this case there is a series
of difficulties related with the impossibility of employing traditional methods and approaches
used for linear equations. At present, a quite topical problem is to determine the conditions
ensuring the uniform convergence of approximate solutions obtained by various methods for
nonlinear equations and system of equations of variational kind. In this case, it is natural
to employ variational methods of solving boundary value problems. And this is an issue on
the justification of these methods arises, which is reduced to studying general properties of
approximate solutions.
We consider the issue on convergence of approximate solutions for the minimal surface equation
$$
\sum_{i=1}^n\frac{\partial}{\partial
x_i}(\frac{f_{x_i}} {\sqrt{1+\nabla f^2}})=0
$$
in domain $\Omega$ subject to the boundary condition
$$
f_{\partial\Omega}=\varphi_{\partial\Omega},
$$
where $\varphi\in C(\overline{\Omega})$. It should be noted that this Dirichlet problem is not solvable for an arbitrary
domain (even with a smooth boundary). For planar domains the necessary and sufficient
condition for the solvability of the Dirichlet problem for an arbitrary continuous boundary
function $\varphi(x)$ is the convexity of this domain. In the space of dimension greater than two,
such condition is the nonnegativity of the mean curvature of the boundary w.r.t. the outward
normal. In the present work we impose no conditions for domain, but we assume that for a given boundary function problem (1)–(2) is solvable. It is clear that for an arbitrary domain , such
functions $\Omega$ exist.
We study the issue on uniform convergence of polynomial approximate solutions to the
minimal surface equation constructed by means of algebraic polynomials. In work [4] a similar
convergence problem for piecewise linear approximate solution to boundary value problem (1)–(2) was solved, while in work [8] there was given a description of numerical realization of finite
elements methods based on piecewise linear functions. Let us provide required definitions.
In what follows we shall be interested in the issue on uniform convergence of a sequence of
polynomial solutions $\varphi+v^*_N$
as $N\to\infty$. First of all we shall show that under certain conditions,
the gradients of these functions are bounded by a constant independent of $N$. This property
will allow us to obtain an estimate for the rate of uniform convergence to the exact solution.
Keywords:
minimal surface equation, uniform convergence, almost solution, approximation of equation, estimation of uniform convergence.
DOI:
https://doi.org/10.15688/jvolsu1.2016.5.11
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Citation:
I. V. Truhlyaeva, “On the convergence of almost polynomial solutions of the minimal surface”, Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2016, no. 5(36), 124–139
Citation in format AMSBIB
\Bibitem{Tru16}
\by I.~V.~Truhlyaeva
\paper On the convergence of almost polynomial solutions of the minimal surface
\jour Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica
\yr 2016
\issue 5(36)
\pages 124139
\mathnet{http://mi.mathnet.ru/vvgum136}
\crossref{https://doi.org/10.15688/jvolsu1.2016.5.11}
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