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Тезисы докладов

Численное решение дифференциальных и интегральных уравнений

The one-dimensional Dirichlet problem for non-self-conjugate elliptic equ-ation of the second order with coordinated degeneration of initial data and with strong singularity of a solution at the origin is considered. For this problem we define the solution as the $R_nu$-generalized one. Such a definition allowed to establish the existence and uniqueness of solution in $H^1_{2,nu+{beta}/2}$ (see [1]). In the present paper it has been proved that solution belongs to the weighted Sobolev space $H^3_{2,nu+{beta}/2+1}$ under proper assumptions on coefficients and the right-hand side of differential equation. The scheme of the finite element method is constructed taking into account a specific character of the problem, namely the strong singularity of its solution. To this end we fix the mesh and introduce the finite element space containing singular polynomials (see [2]). The increase of dimension of the finite element space is achieved by the growth of degrees of approximating polynomials on finite elements. That is why we call this approach as the $p$-version of the finite element method for the boundary value problem with strong singularity of a solution. Using the established regularity of an $R_nu$-generalized solution to the problem, the estimate of the second order for the error of approximation has been obtained in the norm of the weighted Sobolev space. Besides that, for the problem under consideration it has been established that the $p$-version of the finite element method has the rate of convergence which is the double of the rate for the $h$-version with quasiuniform mesh.

Примечание. Тезисы докладов публикуются в авторской редакции

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