Newton additive and multiplicative Schwarz iterative methods

Convergence properties are presented for Newton additive and multiplicative Schwarz (AS and MS) iterative methods for the solution of nonlinear systems in several variables. These methods consist of approximate solutions of the linear Newton step using either AS or MS iterations, where overlap between subdomains can be used. Restricted versions of these methods are also considered. These Schwarz methods can also be used to precondition a Krylov subspace method for the solution of the linear Newton steps. Numerical experiments on parallel computers are presented, indicating the effectiveness of these methods.The Spanish Ministry of Science and Education (TIN2005-09037-C02-02); Universidad de Alicante (VIGROB-020); the U.S. Department of Energy (DE-FG02-05ER25672)

Stable multilevel splittings of boundary edge element spaces

We establish the stability of nodal multilevel decompositions of lowest-order conforming boundary element subspaces of the trace space ${\boldsymbol{H}}^{-\frac {1}{2}}(\operatorname {div}_{\varGamma },{\varGamma })$ of ${\boldsymbol{H}}(\operatorname {\bf curl},{\varOmega })$ on boundaries of triangulated Lipschitz polyhedra. The decompositions are based on nested triangular meshes created by uniform refinement and the stability bounds are uniform in the number of refinement levels. The main tool is the general theory of P.Oswald (Interface preconditioners and multilevel extension operators, in Proc. 11th Intern. Conf. on Domain Decomposition Methods, London, 1998, pp.96-103) that teaches, when stability of decompositions of boundary element spaces with respect to trace norms can be inferred from corresponding stability results for finite element spaces. ${\boldsymbol{H}}(\operatorname {\bf curl},{\varOmega })$ -stable discrete extension operators are instrumental in this. Stable multilevel decompositions immediately spawn subspace correction preconditioners whose performance will not degrade on very fine surface meshes. Thus, the results of this article demonstrate how to construct optimal iterative solvers for the linear systems of equations arising from the Galerkin edge element discretization of boundary integral equations for eddy current problem

Stationary splitting iterative methods for the matrix equation AX B = C

Stationary splitting iterative methods for solving AXB = Care considered in this paper. The main tool to derive our new method is the induced splitting of a given nonsingular matrix A = M −N by a matrix H such that (I −H) invertible. Convergence properties of the proposed method are discussed and numerical experiments are presented to illustrate its computational efficiency and the effectiveness of some preconditioned variants. In particular, for certain surface fitting applications, our method is much more efficient than the progressive iterative approximation (PIA), a conventional iterative method often used in computer-aided geometric design (CAGD).The authors would like to thank the supports of the National Natural Science Foundation of China under Grant No. 11371075, the Hunan Key Laboratory of mathematical modeling and analysis in engineering, and the Portuguese Funds through FCT–Fundação para a Ciência e a Tecnologia, within the Project UID/MAT/00013/2013

The shifted classical circulant and skew circulant splitting iterative methods for Toeplitz matrices

It is known that every Toeplitz matrix T enjoys a circulant and skew circulant splitting (denoted by CSCS), i.e., T=C-S with C a circulant matrix and S a skew circulant matrix. Based on the variant of such a splitting (also referred to as CSCS), we first develop classical CSCS iterative methods and then introduce shifted CSCS iterative methods for solving hermitian positive definite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical CSCS iterative methods work slightly better than the Gauss-Seidel (GS) iterative methods if the CSCS is convergent and that there is always a constant $\alpha$ such that the shifted CSCS iteration converges much faster than the Gauss-Seidel iteration, no matter whether the CSCS itself is convergent or not.National Natural Science Foundation of China No. 11371075The authors would like to thank the supports of the National Natural Science Foundation of China under Grant No. 11371075, the research innovation program of Hunan province for postgraduate students under Grant No. CX2015B374, the Portuguese Funds through FCT–Fundac˜ao para a Ciˆencia, within the Project UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Some comparison theorems for weak nonnegative splittings of bounded operators

AbstractThe comparison of the asymptotic rates of convergence of two iteration matrices induced by two splittings of the same matrix has arisen in the works of many authors. In this paper we derive new comparison theorems for weak nonnegative splittings and weak splittings of bounded operators in a general Banach space and rather general cones, and in a Hilbert space, which extend some of the results obtained by Woźnicki (Japan J. Indust. Appl. Math. 11(1994) 289–342) and Marek and Szyld (Numer. Math. 44(1984) 23–35). Furthermore, we present new theorems also for bounded operator which extend some results by Csordas and Varga (Numer. Math. 44. (1984) 23–35) for weak nonnegative splittings of matrices

Some comparison theorems for weak nonnegative splittings of bounded operators

AbstractThe comparison of the asymptotic rates of convergence of two iteration matrices induced by two splittings of the same matrix has arisen in the works of many authors. In this paper we derive new comparison theorems for weak nonnegative splittings and weak splittings of bounded operators in a general Banach space and rather general cones, and in a Hilbert space, which extend some of the results obtained by Woźnicki (Japan J. Indust. Appl. Math. 11(1994) 289–342) and Marek and Szyld (Numer. Math. 44(1984) 23–35). Furthermore, we present new theorems also for bounded operator which extend some results by Csordas and Varga (Numer. Math. 44. (1984) 23–35) for weak nonnegative splittings of matrices

Multigrid Waveform Relaxation on Spatial Finite Element Meshes: The Discrete-Time Case

The efficiency of numerically solving time-dependent partial differential equations on parallel computers can be greatly improved by computing the solution on many time levels simultaneously. The theoretical properties of one such method, namely the discrete-time multigrid waveform relaxation method, are investigated for systems of ordinary differential equations obtained by spatial finite-element discretisation of linear parabolic initial-boundary value problems. The results are compared to the corresponding continuous-time results. The theory is illustrated for a one-dimensional and a two-dimensional model problem and checked against results obtained by numerical experiments