On convergence of polynomial approximate solutions of minimal surface equations in domains satisfying the cone condition

I. V. Truhlyaeva

Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 23 (2020) no. 4, pp. 5-12.

Résumé

In this paper we consider the polynomial approximate solutions of the 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 systems of equations of variational kind (see, for instance, [1]). For nonlinear equations it is first necessary to establish some a priori estimates of the derivatives of approximate solutions. In this paper, we gave a substantiation of the variational method of solving the minimal surface equation in the case of multidimensional space. We use the same approach that we used in [3] for a two-dimensional equation. Note that such a convergence was established in [3] under the condition that a certain geometric characteristic $\Delta(\Omega)$ in the domain $\Omega$, in which the solutions are considered, is positive. In particular, domain with a smooth boundary satisfied this requirement. However, this characteristic is equal to zero for a fairly wide class of domains with piecewise-smooth boundaries and sufficiently“narrow” sections at the boundary. For example, such a section of the boundary is the vertex of a cone with an angle less than $\pi$/2. In this paper, we present another approach to determining the value of $\Delta(\Omega)$ in terms of which it is possibleto extend the results of the work [3] in domains satisfying the cone condition.

Détail

Keywords: minimal surface equation, approximate solution, approximation of equations
Mots-clés : uniform convergence, estimation of uniform convergence.