Robin Approximation of Dirichlet Boundary Value Problems

Abstract

ABSTRACTThis paper describes the rate of convergence of solutions of Robin boundary value problems of an elliptic equation to the solution of a Dirichlet problem as a boundary parameter decreases to zero. The results are found using representations for solutions of the equations in terms of Steklov eigenfunctions. Particular interest is in the case where the Dirichlet data is only in L2(∂Ω,dσ). Various approximation bounds are obtained and the rate of convergence of the Robin approximations in the H1 and L2 norms are shown to have convergence rates that depend on the regularity of the Dirichlet data.