Efficient solution of nonlinear elliptic problems using hierarchical matrices with Broyden updates

Mario Bebendorf and Ying Chen

Contact the author: Please use for correspondence this email.
Submission date: 27. May. 2005
Pages: 17
published in: Computing, 81 (2007) 4, p. 239-257 
DOI number (of the published article): 10.1007/s00607-007-0252-0
Bibtex
MSC-Numbers: 34B15, 90C53, 65N30, 65F50
Keywords and phrases: quasi-newton methods, broyden updates, nonlinear elliptic equations
Download full preprint: PDF (194 kB), PS ziped (192 kB)

Abstract:
The numerical solution of nonlinear problems is usually connected with Newton's method. Due to its computational cost, variants (so-called quasi-Newton methods) have been developed in which the arising inverse of the Jacobian is replaced by an approximation. In this article we present a new approach which is based on Broyden updates. Instead of updating the inverse we introduce a method which constructs updates of the LU decomposition. Since an approximate LU decomposition of finite element stiffness matrices can be efficiently computed in the set of hierarchical matrices, the complexity of the proposed method scales almost linearly. Numerical examples demonstrate the effectiveness of this new approach.