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Modelling and Simulation in Engineering
Volume 2019, Article ID 5157145, 9 pages
Research Article

The Numerical Solution of Singularly Perturbed Nonlinear Partial Differential Equations in Three Space Variables: The Adaptive Explicit Inverse Preconditioning Approach

Department of Informatics and Telematics, Harokopio University, Athens, Greece

Correspondence should be addressed to Anastasia-Dimitra Lipitakis; moc.liamg@ikatipilda

Received 30 August 2018; Accepted 4 December 2018; Published 2 January 2019

Academic Editor: Michele Calì

Copyright © 2019 Anastasia-Dimitra Lipitakis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Critical comments on the complexity of computational systems and the basic singularly perturbed (SP) concepts are given. A class of several complex SP nonlinear elliptic equations arising in various branches of science, technology, and engineering is presented. A classification of complex SP nonlinear PDEs with characteristic boundary value problems is described. A modified explicit preconditioned conjugate gradient method based on explicit inverse preconditioners is presented. The numerical solution of a characteristic 3D SP nonlinear parabolic model is analytically given and numerical results for several model problems are presented demonstrating both applicability and efficiency of the new computational methods.