Communications in Mathematical Sciences

Volume 10 (2012)

Number 2

Numerical solution of bi-periodic elliptic problems in unbounded domains

Pages: 513 – 526



Chunxiong Zheng (Department of Mathematical Sciences, Tsinghua University, Beijing, China)


This paper aims at an efficient numerical approach for bi-periodic elliptic problems with local defects in unbounded domains. We employ the methodology of artificial boundary methods and try to design an accurate boundary condition in the form of a Dirichlet-to-Neumann (DtN) map. The key issue is how to take advantage of periodicity as much as possible. We develop an approach of computing the DtN map based on the DtN gluing and homogenization techniques, and prove the unique solvability of the resulting discrete variational problem. Numerical evidence validates the effectiveness of the proposed method.


periodic structure, DtN homogenization, DtN gluing, artificial boundary method

2010 Mathematics Subject Classification

35J50, 35J56

Published 21 December 2011