Dans cet article nous donnons une formule pour les coefficients de l’inverse des matrices de Toeplitz respectivement de symboles $f\left(e^{i\theta }\right)=(1-cos\theta ){\left|f_1\left(e^{i\theta }\right)\right|}^2$ (cas singulier) et $|f_1\left(e^{i\theta }\right){|}^2$ (cas régulier) où $f_1$ est une fonction appartenant à  une classe de fonctions holomorphes sur un disque ouvert contenant le tore $𝕋$ et sans zéro sur $𝕋$. Un cas particulier défini par $f_1=\frac{Q}{P}$ où $P$ et $Q$ sont des polynômes sans zéro sur $𝕋$ est traité. Dans le cas où le symbole est singulier, cette formule présente l’intérêt d’avoir un second ordre. Dans tous les...

For the efficient numerical solution of indefinite linear systems arising from curl conforming edge element approximations of the time-harmonic Maxwell equation, we consider local multigrid methods (LMM) on adaptively refined meshes. The edge element discretization is done by the lowest order edge elements of Nédélec’s first family. The LMM features local hybrid Hiptmair smoothers of Jacobi and Gauss–Seidel type which are performed only on basis functions associated with newly created edges/nodal...

For the efficient numerical solution of indefinite linear systems arising from curl conforming edge element approximations of the time-harmonic Maxwell equation, we consider local multigrid methods (LMM) on adaptively refined meshes. The edge element discretization is done by the lowest order edge elements of Nédélec’s first family. The LMM features local hybrid Hiptmair smoothers of Jacobi and Gauss–Seidel type which are performed only on basis functions associated with newly created edges/nodal...

In this paper, our attention is concentrated on the GMRES method for the solution of the system $(I-T)x=b$ of linear algebraic equations with a nonsymmetric matrix. We perform $m$ pre-iterations $y_{l+1}=T{y}_l+b$ before starting GMRES and put $y_m$ for the initial approximation in GMRES. We derive an upper estimate for the norm of the error vector in dependence on the $m$th powers of eigenvalues of the matrix $T$. Further we study under what eigenvalues lay-out this upper estimate is the best one. The estimate shows and numerical...