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Volume 10, Issue 2
Convergence of Rothe Scheme for Hemivariational Inequalities of Parabolic Type

P. Kalita

Int. J. Numer. Anal. Mod., 10 (2013), pp. 445-465.

Published online: 2013-10

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  • Abstract

This article presents the convergence analysis of a sequence of piecewise constant and piecewise linear functions obtained by the Rothe method to the solution of the first order evolution partial differential inclusion $u'(t)+Au(t)+\iota^{*} ∂ J(\iota u(t)) \ni f(t)$, where the multivalued term is given by the Clarke subdifferential of a locally Lipschitz functional. The method provides the proof of existence of solutions alternative to the ones known in literature and together with any method for underlying elliptic problem, can serve as the effective tool to approximate the solution numerically. Presented approach puts into the unified framework known results for multivalued nonmonotone source term and boundary conditions, and generalizes them to the case where the multivalued term is defined on the arbitrary reflexive Banach space as long as appropriate conditions are satisfied. In addition, the results on improved convergence as well as the numerical examples are presented.

  • AMS Subject Headings

35K86, 47J22, 65M12

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COPYRIGHT: © Global Science Press

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@Article{IJNAM-10-445, author = {}, title = {Convergence of Rothe Scheme for Hemivariational Inequalities of Parabolic Type}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2013}, volume = {10}, number = {2}, pages = {445--465}, abstract = {

This article presents the convergence analysis of a sequence of piecewise constant and piecewise linear functions obtained by the Rothe method to the solution of the first order evolution partial differential inclusion $u'(t)+Au(t)+\iota^{*} ∂ J(\iota u(t)) \ni f(t)$, where the multivalued term is given by the Clarke subdifferential of a locally Lipschitz functional. The method provides the proof of existence of solutions alternative to the ones known in literature and together with any method for underlying elliptic problem, can serve as the effective tool to approximate the solution numerically. Presented approach puts into the unified framework known results for multivalued nonmonotone source term and boundary conditions, and generalizes them to the case where the multivalued term is defined on the arbitrary reflexive Banach space as long as appropriate conditions are satisfied. In addition, the results on improved convergence as well as the numerical examples are presented.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/577.html} }
TY - JOUR T1 - Convergence of Rothe Scheme for Hemivariational Inequalities of Parabolic Type JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 445 EP - 465 PY - 2013 DA - 2013/10 SN - 10 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/577.html KW - hemivariational inequality, Rothe method, convergence, existence. AB -

This article presents the convergence analysis of a sequence of piecewise constant and piecewise linear functions obtained by the Rothe method to the solution of the first order evolution partial differential inclusion $u'(t)+Au(t)+\iota^{*} ∂ J(\iota u(t)) \ni f(t)$, where the multivalued term is given by the Clarke subdifferential of a locally Lipschitz functional. The method provides the proof of existence of solutions alternative to the ones known in literature and together with any method for underlying elliptic problem, can serve as the effective tool to approximate the solution numerically. Presented approach puts into the unified framework known results for multivalued nonmonotone source term and boundary conditions, and generalizes them to the case where the multivalued term is defined on the arbitrary reflexive Banach space as long as appropriate conditions are satisfied. In addition, the results on improved convergence as well as the numerical examples are presented.

P. Kalita. (1970). Convergence of Rothe Scheme for Hemivariational Inequalities of Parabolic Type. International Journal of Numerical Analysis and Modeling. 10 (2). 445-465. doi:
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