Analysis of Dirichlet\u2013Robin Iterations for Solving the Cauchy Problem for Elliptic Equations

Pauline Achieng,Vladimir Kozlov,Fredrik Berntsson,Jennifer Chepkorir

doi:10.1007/s41980-020-00466-7

Abstract

The Cauchy problem for general elliptic equations of second order is considered. In a previous paper (Berntsson et al. in Inverse Probl Sci Eng 26(7):1062–1078, 2018), it was suggested that the alternating iterative algorithm suggested by Kozlov and Maz’ya can be convergent, even for large wavenumbers k^2, in the Helmholtz equation, if the Neumann boundary conditions are replaced by Robin conditions. In this paper, we provide a proof that shows that the Dirichlet–Robin alternating algorithm is indeed convergent for general elliptic operators provided that the parameters in the Robin conditions are chosen appropriately. We also give numerical experiments intended to investigate the precise behaviour of the algorithm for different values of k^2 in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

Highlights

Let be a bounded domain in Rd with a Lipschitz boundary divided into two disjoint parts 0 and 1 such that they have a common Lipschitz boundary in and 0 ∪ 1 =, see Fig. 1
Under the assumption that the elliptic operator with the Dirichlet boundary condition is positive we show that the Dirichlet–Robin algorithm is convergent, provided that parameters in the Robin conditions are chosen appropriately
The results show that for small values of μ, the Dirichlet–Robin algorithm is divergent but for sufficiently large values of μ, we obtain convergence

Summary

Introduction

Νd ) is the outward unit normal to the boundary , D j = ∂/∂x j , a ji and a are measurable real-valued functions such that a is bounded, ai j = a ji and λ|ξ |2 ≤ ai j ξi ξ j ≤ λ−1|ξ |2, ξ ∈ Rd , λ = const > 0. We are seeking real-valued solutions to problem (1.1). We will always assume that there is only trivial solution to Lu = 0 in H 1( ) if u = 0, N u = 0 on 0 or u = 0, N u = 0 on 1. This is certainly true for the Helmholtz equation

Objectives

Results

Conclusion