A class of preconditioned conjugate gradient methods applied to finite element equations
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It is well known that certain conditions have to be fulfilled to ensure existence of stable incomplete (IC) factorizations of a matrix.
Under additional conditions it is possible to construct modified incomplete (MIC) factorizations such that the order of convergence of the corresponding preconditioned conjugate gradient method is increased. For finite element discretizations of partial differential equations leading to matrices meeting these conditions, the number of iterations is of order O(h−1/2), h → 0, compared to O(h−1) for the standard incomplete (IC) factorizations. Here, h is a mesh-parameter.
Since particularly one of the conditions, the so called MIC-condition, sometimes is violated in applications, while the IC-conditions are fulfilled, this point of the theory will be stressed. The simplest method, the so called MIC(0)*-method, will be used as a model in this context.
In this survey, methods for satisfying the fundamental conditions, when they are not primarily fulfilled, will be discussed. Two such ideas are spectral equivalence and hierarchical finite element techniques. A number of applications, e g from plane strain linear elasticity problems, where these ideas are used, will be presented.
For certain model-problems, it is possible to construct modified incomplete factorizations based only on element-matrix factors, such that the actual increase of the rate of convergence is still obtained. This idea will be briefly discussed too.
Key WordsFinite element equations preconditioned iterative methods conjugate gradients incomplete factorization
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