Convergence of the finite element method for boundary value problem with degeneration on the whole boundary of domain

E. I. Rukavishnikova

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ Vyčislitelʹnaâ matematika i informatika, Tome 8 (2019) no. 3, pp. 5-26

Résumé

In this paper we consider the Dirichlet problem with homogeneous boundary condition for a second-order elliptic equation with degeneration on the entire twice continuously differentiable boundary of two-dimensional domain $\Omega$. We define a generalized solution of this problem which exists and is unique in the weighted Sobolev space $\mathring{W}^1_{2,\alpha}(\Omega)$. To solve the formulated problem a finite element method is developed, the scheme of which is constructed on the basis of the definition of a generalized solution of the original differential problem in the space $\mathring{W}^1_{2,\alpha}(\Omega)$. For this purpose a two-dimensional convex domain is divided into triangles with special condensation to the boundary. Next we introduce a finite element space $V^h\subset\mathring{W}^1_{2,\alpha}(\Omega)$ that contains continuous functions liner on each triangular element of grid region $\Omega^h$ and equal to zero on the set $\bar{\Omega}\setminus\Omega^h$, show unique solvability of the scheme of the finite element method. For the generalized solution $u$ from the subspace $\mathring{W}^2_{2,\alpha-1}(\Omega)$ of the space $\mathring{W}^1_{2,\alpha}(\Omega)$, using its values in the nodes of the triangulated domain, an interpolant $u_I\in V^h$ is constructed, the fact of its convergence with respect to the norm $W^1_{2,\alpha}(\Omega)$ is established. The main result of the work for the proposed method for solving the first boundary value problem with degeneration is the proof of the convergence of the approximate solution to the exact solution in the weighted Sobolev space.

Détail

Keywords: boundary value problem with degeneration, Sobolev weighted space, generalized solution, finite element method.