About sampling of the two-dimensional Laplace operator in a smooth two-dimensional area

S. D. Algazin

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 24 (2021) no. 3, pp. 253-259

Résumé

Currently the finite element method is most widespread technique for solving problems of mechanics of a deformable solid body. Its shortcomings are well-known: approximating a displacement by a piecewise-linear function, we obtain the tension to be discontinuous. At the same time, it is necessary to notice that most problems of mechanics of a deformable solid body are described by the elliptic type equations which have smooth decisions. It seems to be relevant to develop algorithms which would take this smoothness into account. The idea of such algorithms belongs to K.I. Babenko. This idea was stated in the early seventies of the last century. A long-lasting application of this technique in elliptic tasks to eigenvalues has proved their high performance to the author of this study. However, in this technique the matrix of the finite-dimensional task turns out to be not symmetric but only close to that to be symmetrized. Below, the application when sampling the Bubnov-Galyorkina method, this defect is eliminated. Let us note that the symmetry of the matrix of the finite-dimensional task is important when studying the stability. Unlike classical difference methods and the finite element method where the dependence of the convergence ratio on the number of nodes of the grid is power, we have an exponential decrease of the error.