Preconditioning isoparametric finite element methods taking into account numerical integration (Q1126613)

This paper is concerned with the \(H^1\)-condition numbers and the \(B_h\)-singular value distribution of preconditioned operators \(\{ B^{-1}_h A_h \}_{0 < h < 1}\), where \(A_h\) and \(B_h\) are isoparametric finite element discretizations of second-order elliptic operators \(A\) and \(B_1\) by taking into account numerical integration respectively, where \(A\) is nonselfadjoint and possibly indefinite, and \(B\) is selfadjoint and positive definite. It is proved that the \(H^1\)-condition numbers of operators \(\{ B^{-1}_h A_h \}_{0 < h < 1}\) are uniformly bounded, and the \(B_h\)-singular values cluster in a positive finite interval. Finally, the implementation of \(B^{-1}_h\) is discussed.