# Investigation of the dependence of the error in the approximate solution of the Laplace equation on the mean minimum sine of the angle of the cells of the computational grid

Abstract

## Investigation of the dependence of the error in the approximate solution of the Laplace equation on the mean minimum sine of the angle of the cells of the computational grid

Incoming article date: 17.12.2021The paper studies the issue of the influence of the quality of the computational triangular grid on the accuracy of calculations in various computational problems. There is a well-known example of Schwartz, which shows that the approximation of a smooth surface by a polyhedral surface can give very large errors for calculating the surface area. This is due to the quality of the constructed triangulation of the surface. Therefore, it is natural to expect that there is some connection between a certain triangulation characteristic and the accuracy of solving some computational problem. In the presented article, as such a characteristic, a value is chosen - the average value of the minimum sine of the angle of all triangles of the computational grid. In the course of numerical experiments, the Dirichlet problem for the Laplace equation in a circular ring was solved, in which the error of the approximate solution was calculated (the gradient descent method was used to find a solution to the corresponding variational problem.). For the ring, a series of triangulations was constructed with a uniform division along the angle and a non-uniform division along the radius in polar coordinates. In this example, a linear dependence of the error on was shown. The article presents both the results of the calculation with different values and the calculation of the correlation coefficient of the studied quantities.

Keywords:boundary value problem, Delaunay triangulation, calculation accuracy, Dirichlet problem, mathematical modeling, triangular mesh, minimum triangle angle, piecewise linear approximation, variational method, Laplace equation