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Methods for overcoming breakdown problems in the Unsymmetric Lanczos Reduction method
Author(s) -
Li Henian,
Aitchison Peter,
Woodbury Allan
Publication year - 1998
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/(sici)1097-0207(19980615)42:3<389::aid-nme362>3.0.co;2-9
Subject(s) - lanczos resampling , reduction (mathematics) , lanczos algorithm , mathematical optimization , finite element method , instability , computer science , mathematics , algorithm , engineering , structural engineering , mechanics , physics , geometry , eigenvalues and eigenvectors , quantum mechanics
The Unsymmetric Lanczos Reduction (ULR) method is developed to solve the finite‐element‐based solution to the contaminant transport problem. The method sometimes suffers from breakdown when at some step division by a pivot which is zero or near zero, causes numerical instability. In this paper, the Maximum‐Pivot New‐Start Vector method is developed to overcome such breakdowns by constructing a new starting vector with the possible maximum pivot. Some cases of instability cannot be remedied by this approach (pathological breakdowns) and the Switch method is developed to complete the solution by changing the algorithm to an Arnoldi reduction approach. Investigation of some two‐dimensional examples and field problems illustrates the efficiency of the methods and substantial time savings over other existing solution methods. © 1998 John Wiley & Sons, Ltd.