Construction of a minimal parametrically specified surface by the method of minimizing the Dirichlet integral

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. In the course of this work, an algorithm for numerical modeling of minimal surfaces was developed based on the Weierstrass-Enneper representation. An algorithm was also implemented in Python that allows you to calculate approximate minimal surfaces in the class of polynomial vector functions defined on the unit disk and satisfying the boundary condition Dirichlet (Plateau problem).