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A review of reliable numerical models for three‐dimensional linear parabolic problems
Author(s) -
Faragó I.,
Horváth R.
Publication year - 2006
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/nme.1863
Subject(s) - thermal conduction , finite element method , partial differential equation , mathematics , a priori and a posteriori , norm (philosophy) , parabolic partial differential equation , finite difference , heat equation , mathematical optimization , mathematical analysis , thermodynamics , physics , philosophy , epistemology , political science , law
The preservation of characteristic qualitative properties of different phenomena is a more and more important requirement in the construction of reliable numerical models. For phenomena that can be mathematically described by linear partial differential equations of parabolic type (such as the heat conduction, the diffusion, the pricing of options, etc.), the most important qualitative properties are: the maximum–minimum principle, the non‐negativity preservation and the maximum norm contractivity. In this paper, we analyse the discrete analogues of the above properties for finite difference and finite element models, and we give a systematic overview of conditions that guarantee the required properties a priori . We have chosen the heat conduction process to illustrate the main concepts, but engineers and scientists involved in scientific computing can easily reformulate the results for other problems too. Copyright © 2006 John Wiley & Sons, Ltd.