The Finite Element Method of High Degree of Accuracy for Boundary Value Problem with Singularity

Abstract

Mathematical models of fracture physics and mechanics are boundary value problems for differential equations and systems of equations with a singularity. There are two classes of problems with a singularity: with coordinated and uncoordinated degeneracy of the input data, depending on the behavior of the coefficients of the equation. Finite element methods with the first order of convergence rate O(h) have been created to find an approximate solution to these problems. We construct a scheme of the weighted finite element method of high degree of accuracy for the boundary value problem with uncoordinated degeneracy of the input data and singularity of the solution. The rate of convergence of an approximate solution of the proposed finite element method to the exact Rν-generalized solution in the weight set W2,ν+β2+21(Ω,δ) is investigated. The estimation of finite element approximation O(h2) is established.