On the convergence of basic iterative methods for convection-diffusion equations (Q2760360)

Let us denote by \(PD\) ad \(SPD\) the class of positive definite and symmetric positive definite matrices respectively. Let \(M_0\) be the class of matrices from \(PD\) with non-positive off-diagonal elements and let \(SPD.M_0\) denote the class of matrices \(A=A_d + A_c\) such that \(A_d \in SPD\), \(A_c \in M_0\). NEWLINENEWLINENEWLINEThe authors prove that the damped Jacobi iteration converges for systems of linear equations with matrices from the superclass \(PD\) of the class \(SPD.M_0\) and describe the related damping parameters \(\Theta \). They present matrices from the class \(SPD.M_0\) for which the damped Gauss-Seidel iteration diverges for all values of \(\Theta \). Finally, they prove that a modification of the damped Gauss-Seidel method, called a hybrid method, converges in the whole class \(PD\). NEWLINENEWLINENEWLINEThese results are shown to be useful in the study of convergence of the above-mentioned iteration methods for systems of linear equations resulting from finite element discretizations of the convection-diffusion problems by means of the artificial diffusion method, the Tabata upwind triangle method and by the box method.