Domain Decomposition–Finite Difference Approximate Inverse Preconditioned Schemes for Solving Fourth-Order Equations

A new class of explicit approximate inverse preconditioning is introduced for solving fourth-order equations, based on the ‘coupled equation approach’, by the domain decomposition method in conjunction with various finite difference approximation schemes. Explicit approximate inverse arrow-type matrix techniques, based on the concept of sparse LU-type factorization procedures, are introduced for computing a class of approximate inverses. Explicit preconditioned conjugate gradient-type schemes are presented for the efficient solution of linear systems. Applications of the method to a biharmonic problem are discussed and numerical results are given.

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