On the Error Estimation of the Finite Element Method for the Boundary Value Problems with Singularity in the Lebesgue Weighted Space

Abstract

Abstract A boundary value problem is said to possess a strong singularity if its solution u does not belong to the Sobolev space or, in other words, the Dirichlet integral of the solution u diverges. This article analyzes the finite element method for the boundary value problems with coordinated and uncoordinated degeneration of input data and with strong singularity of the solution. The scheme of the finite element method is constructed on the basis of the definition of R ν-generalized solution to the problem, and the finite element space contains singular power functions. It is established that the considered method has second order convergence to the exact R ν generalized solution in the norm of the Lebesgue weighted space. Keywords: Finite element method R ν-generalized solutionStrong singularity of solutionMathematics Subject Classification: 65N30 ACKNOWLEDGMENTS The work was supported by the Russian Foundation of Basic Research (10-01-00060, 11-01-98502-east). This article is dedicated to Professor Gennadi Vainikko for his 75th birthday.