Numerical Solution of an Elliptic Problem with a Non-classical Boundary Condition

摘要

We investigate an elliptic problem with a boundary condition given by a sum of normal derivative and an elliptic operator in tangential variables (also known as "Venttsel" boundary condition). The differential problem is discretized by a specific finite difference method. Error estimates of the numerical method in the discrete Sobolev space W_2~1 are obtained. The rate of convergence in this space is optimal, i.e. it is m - 1 for solutions from W_2~m, 1 < m < 2.5.