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Volume 35, Issue 4
A Multigrid Semismooth Newton Method for Semilinear Contact Problems

Michael Ulbrich, Stefan Ulbrich & Daniela Bratzke

DOI: 10.4208/jcm.1702-m2016-0679

J. Comp. Math., 35 (2017), pp. 486-528.

Published online: 2017-08

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  • Abstract

This paper develops and analyzes multigrid semismooth Newton methods for a class of inequality-constrained optimization problems in function space which are motivated by and include linear elastic contact problems of Signorini type. We show that after a suitable Moreau-Yosida type regularization of the problem superlinear local convergence is obtained for a class of semismooth Newton methods. In addition, estimates for the order of the error introduced by the regularization are derived. The main part of the paper is devoted to the analysis of a multilevel preconditioner for the semismooth Newton system. We prove a rigorous bound for the contraction rate of the multigrid cycle which is robust with respect to sufficiently small regularization parameters and the number of grid levels. Moreover, it applies to adaptively refined grids. The paper concludes with numerical results.

  • Keywords

Contact problems, Semismooth Newton methods, Multigrid methods, Error estimates.

  • AMS Subject Headings

65K10, 65C20, 65N30, 65N55, 74M15.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

mulbrich@ma.tum.de (Michael Ulbrich)

ulbrich@mathematik.tu-darmstadt.de (Stefan Ulbrich)

bratzke@mathematik.tu-darmstadt.de (Daniela Bratzke)

  • BibTex
  • RIS
  • TXT
@Article{JCM-35-486, author = {Ulbrich , MichaelUlbrich , Stefan and Bratzke , Daniela}, title = {A Multigrid Semismooth Newton Method for Semilinear Contact Problems}, journal = {Journal of Computational Mathematics}, year = {2017}, volume = {35}, number = {4}, pages = {486--528}, abstract = {

This paper develops and analyzes multigrid semismooth Newton methods for a class of inequality-constrained optimization problems in function space which are motivated by and include linear elastic contact problems of Signorini type. We show that after a suitable Moreau-Yosida type regularization of the problem superlinear local convergence is obtained for a class of semismooth Newton methods. In addition, estimates for the order of the error introduced by the regularization are derived. The main part of the paper is devoted to the analysis of a multilevel preconditioner for the semismooth Newton system. We prove a rigorous bound for the contraction rate of the multigrid cycle which is robust with respect to sufficiently small regularization parameters and the number of grid levels. Moreover, it applies to adaptively refined grids. The paper concludes with numerical results.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1702-m2016-0679}, url = {http://global-sci.org/intro/article_detail/jcm/10028.html} }
TY - JOUR T1 - A Multigrid Semismooth Newton Method for Semilinear Contact Problems AU - Ulbrich , Michael AU - Ulbrich , Stefan AU - Bratzke , Daniela JO - Journal of Computational Mathematics VL - 4 SP - 486 EP - 528 PY - 2017 DA - 2017/08 SN - 35 DO - http://doi.org/10.4208/jcm.1702-m2016-0679 UR - https://global-sci.org/intro/article_detail/jcm/10028.html KW - Contact problems, Semismooth Newton methods, Multigrid methods, Error estimates. AB -

This paper develops and analyzes multigrid semismooth Newton methods for a class of inequality-constrained optimization problems in function space which are motivated by and include linear elastic contact problems of Signorini type. We show that after a suitable Moreau-Yosida type regularization of the problem superlinear local convergence is obtained for a class of semismooth Newton methods. In addition, estimates for the order of the error introduced by the regularization are derived. The main part of the paper is devoted to the analysis of a multilevel preconditioner for the semismooth Newton system. We prove a rigorous bound for the contraction rate of the multigrid cycle which is robust with respect to sufficiently small regularization parameters and the number of grid levels. Moreover, it applies to adaptively refined grids. The paper concludes with numerical results.

Ulbrich , MichaelUlbrich , Stefan and Bratzke , Daniela. (2017). A Multigrid Semismooth Newton Method for Semilinear Contact Problems. Journal of Computational Mathematics. 35 (4). 486-528. doi:10.4208/jcm.1702-m2016-0679
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