The Finite Element Method for Boundary Value Problems with Strong Singularity and Double Singularity

摘要

A boundary value problem is said to possess strong singularity if its solution u does not belong to the Sobolev space W_2~1 (H~1) or, in other words, the Dirichlet integral of the solution u diverges. We consider the boundary value problems with strong singularity and with double singularity caused the discontinuity of coefficients in the equation on the domain with slot and presence of the corners equal 2π on boundary of this domain. The schemes of the finite element method is constructed on the basis of the definition on R_v-generalized solution to these problems, and the finite element space contains singular power functions. The rate of convergence of the approximate solution to the R_v-generalized solution in the norm of the Sobolev weighted space is established and, finally, results of numerical experiments are presented.