On the $p$-version of the finite element method for the boundary value problem with singularity

E. V. Kashuba ; V. A. Rukavishnikov

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 8 (2005) no. 1, pp. 31-42

Résumé

The one-dimensional first-type boundary value problem for the second order differential equation with strong singularity of a solution caused by coordinated degeneration of input data at the origin is considered. For this problem we define the solution as $R_{\nu}$-generalized one. It has been proved that solution belongs to the weighted Sobolev space $H^3_{2,\nu+\beta/2+1}$ under proper assumptions for coefficients and the right-hand side of the differential equation. The scheme of the finite element method is constructed on a fixed mesh using polynomials of an arbitrary degree $p$ (the $p$-version of the finite element method). The finite element space contains singular polynomials. Using the regularity of $R_{\nu}$-generalized solution, the estimate for the rate of convergence of the second order with respect to the degree $p$ of polynomials is proved in the norm of the weighted Sobolev space.