An incomplete factorization technique for positive definite linear systems

Abstract

This paper describes a technique for solving the large sparse symmetric linear systems that arise from the application of finite element methods. The technique combines an incomplete factorization method called the shifted incomplete Cholesky factorization with the method of generalized conjugate gradients. The shifted incomplete Cholesky factorization produces a splitting of the matrix <italic>A</italic> that is dependent upon a parameter <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is shown that if <italic>A</italic> is positive definite, then there is some <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi>α</mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which this splitting is possible and that this splitting is at least as good as the Jacobi splitting. The method is shown to be more efficient on a set of test problems than either direct methods or explicit iteration schemes.