Development of algorithms for constructing two-dimensional optimal boundary-adaptive grids and their software implementation

Introduction. It is noted that the use of adaptive grids in calculations makes it possible to improve the accuracy and efficiency of computational algorithms without increasing the number of nodes. This approach is especially efficient when calculating nonstationary problems. The objective of this study is the development, construction and software implementation of methods for constructing computational two-dimensional optimal boundary-adaptive grids for complex configuration regions while maintaining the specified features of the shape and boundary of the region. The application of such methods contributes to improving the accuracy, efficiency, and cost-effectiveness of computational algorithms.Materials and Methods. The problem of automatic construction of an optimal boundary-adaptive grid in a simply connected region of arbitrary geometry, topologically equivalent to a rectangle, is considered. A solution is obtained for the minimum set of input information: the boundary of the region in the physical plane and the number of points on it are given. The creation of an algorithm and a mesh generation program is based on a model of particle dynamics. This provides determining the trajectories of individual particles and studying the dynamics of their pair interaction in the system under consideration. The interior and border nodes of the grid are separated through using the mask tool, and this makes it possible to determine the speed of movement of nodes, taking into account the specifics of the problem being solved.Results. The developed methods for constructing an optimal boundary-adaptive grid of a complex geometry region provides solving the problem on automatic grid construction in two-dimensional regions of any configuration. To evaluate the results of the algorithm research, a test problem was solved, and the solution stages were visualized. The computational domain of the test problem and the operation of the function for calculating the speed of movement of interior nodes are shown in the form of figures. Visualization confirms the advantage of this meshing method, which separates the border and interior nodes.Discussion and Conclusions. The theoretical and numerical studies results are important both for the investigation of the grids qualitative properties and for the computational grid methods that provide solving numerical modeling problems efficiently and with high accuracy.

Optimal boundary regularity for some singular Monge-Ampère equations on bounded convex domains

<p style='text-indent:20px;'>By constructing explicit supersolutions, we obtain the optimal global Hölder regularity for several singular Monge-Ampère equations on general bounded open convex domains including those related to complete affine hyperbolic spheres, and proper affine hyperspheres. Our analysis reveals that certain singular-looking equations, such as <inline-formula><tex-math id="M1">\begin{document}$ \det D^2 u = |u|^{-n-2-k} (x\cdot Du -u)^{-k} $\end{document}</tex-math></inline-formula> with zero boundary data, have unexpected degenerate nature.</p>