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A space decomposition method for parabolic equations
Author(s) -
Tai XueCheng
Publication year - 1998
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/(sici)1098-2426(199801)14:1<27::aid-num2>3.0.co;2-n
Subject(s) - domain decomposition methods , mathematics , parabolic partial differential equation , multigrid method , partial differential equation , decomposition method (queueing theory) , convergence (economics) , backward euler method , mathematical analysis , decomposition , space (punctuation) , euler method , domain (mathematical analysis) , schwarz alternating method , finite element method , iterative method , euler equations , algorithm , computer science , discrete mathematics , ecology , physics , biology , economics , thermodynamics , economic growth , operating system
A convergence proof is given for an abstract parabolic equation using general space decomposition techniques. The space decomposition technique may be a domain decomposition method, a multilevel method, or a multigrid method. It is shown that if the Euler or Crank–Nicolson scheme is used for the parabolic equation, then by suitably choosing the space decomposition, only O (| log τ |) steps of iteration at each time level are needed, where τ is the time‐step size. Applications to overlapping domain decomposition and to a two‐level method are given for a second‐order parabolic equation. The analysis shows that only a one‐element overlap is needed. Discussions about iterative and noniterative methods for parabolic equations are presented. A method that combines the two approaches and utilizes some of the good properties of the two approaches is tested numerically. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 27–46, 1998