Body of Optimal Parameters in the Weighted Finite Element Method for the Crack Problem

Abstract

In this paper, a high-accuracy weighted finite element method is constructed and investigated for finding an approximate solution of the crack problem. We consider an approximation of the Lame system in the domain with the reentrant corner 2π at the boundary. A new concept of definition of the solution of the problem is introduced. It allows us to suppress the influence of the singularity on the accuracy of finding an approximate solution, in contrast to the classical approach. We have introduced a weight function into the basis of the finite element method. The accuracy of finding an approximate solution by the weighted finite element method depends on three input parameters. We created an algorithm and establish the body of optimal parameters in the weighted finite element method for the crack problem. The choice of parameters from this set allows us to accurately and stability find an approximate solution with the smallest deviation from the best error. This is required to generate industrial codes.