TY - JOUR
T1 - Local Multilevel Methods for Second-Order Elliptic Problems with Highly Discontinuous Coefficients
JO - Journal of Computational Mathematics
VL - 3
SP - 223
EP - 248
PY - 2012
DA - 2012/06
SN - 30
DO - http://doi.org/10.4208/jcm.1109-m3401
UR - https://global-sci.org/intro/article_detail/jcm/8427.html
KW - Local multilevel method, Adaptive finite element method, Preconditioned conjugate gradient method, Discontinuous coefficients.
AB - <p style="text-align: justify;">In this paper, local multiplicative and additive multilevel methods on adaptively refined meshes are considered for second-order elliptic problems with highly discontinuous coefficients. For the multilevel-preconditioned system, we study the distribution of its spectrum by using the abstract Schwarz theory. It is proved that, except for a few small eigenvalues, the spectrum of the preconditioned system is bounded quasi-uniformly with respect to the jumps of the coefficient and the mesh sizes. The convergence rate of multilevel-preconditioned conjugate gradient methods is shown to be quasi-optimal regarding the jumps and the meshes. Numerical experiments are presented to illustrate the theoretical findings.</p>