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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
51/2005

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

Mario Bebendorf and Ying Chen

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.

Received:
May 27, 2005
Published:
May 27, 2005
MSC Codes:
34B15, 90C53, 65N30, 65F50
Keywords:
quasi-newton methods, broyden updates, nonlinear elliptic equations

Related publications

inJournal
2007 Repository Open Access
Mario Bebendorf and Ying Chen

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

In: Computing, 81 (2007) 4, pp. 239-257