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The Neumann–Neumann method is a commonly employed domain decomposition method for linear elliptic equations. However, the method exhibits slow convergence when applied to semilinear equations and does not seem to converge at all for certain quasilinear equations. We therefore propose two modified Neumann–Neumann methods that have better convergence properties and require fewer computations. We provide numerical results that show the advantages of these methods when applied to both semilinear and quasilinear equations. We also prove linear convergence with mesh-independent error reduction under certain assumptions on the equation. The analysis is carried out on general Lipschitz domains and relies on the theory of nonlinear Steklov–Poincaré operators. [ABSTRACT FROM AUTHOR]

The paper is concerned with the derivation and analysis of the optimized Schwarz type method for heterogeneous, anisotropic diffusion problems discretized by the finite element cell-centered (FECC) scheme. Differently from the standard finite element method (FEM), the FECC method involves only cell unknowns and satisfies local conservation of fluxes by using a technique of dual mesh and multipoint flux approximations to construct the discrete gradient operator. Consequently, if the domain is decomposed into nonoverlapping subdomains, the transmission conditions (on the interfaces between subdomains) associated with the FECC scheme are different from those of the standard FEM. We derive discrete Robin-type transmission conditions in the framework of FECC discretization, which include both weak and strong forms of the Robin terms due to the construction of the FECC's discrete gradient operator. Convergence of the associated iterative algorithm for a decomposition into strip-shaped subdomains is rigorously proved. Two dimensional numerical results for both isotropic and anisotropic diffusion tensors with large jumps in the coefficients are presented to illustrate the performance of the proposed methods with optimized Robin parameters. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we demonstrate a local convergence of an adaptive scalar solver which is practical for strongly diagonal dominant Jacobian problems such as in some systems of nonlinear equations arising from the application of a nonoverlapping domain decomposition method. The method is tested to a nonlinear interface problem of a multichip heat conduction problem. The numerical results show that the method performs slightly better than a Newton–Krylov method. [Copyright &y& Elsevier]

Abstract: Two different interface predictions are considered, in this work, for the solution of multi-dimensional parabolic problem defined over nonoverlapping subdomains. These interface boundary conditions are predicted by the explicit prediction (EP) and the time lagging (TL) methods with δx and 2δx mesh spacings away from the interface lines. We considered the additive splitting up method with respect to the spatial variable to solve the multi-dimensional parabolic problem over each subdomains. For more accurate solution we present the correction for the explicit interface prediction and implement the proposed methods for the solution of two different model problems. [Copyright &y& Elsevier]

This paper deals with discrete monotone iterative algorithms for solving a nonlinear singularly perturbed reaction–diffusion problem. Firstly, the monotone method (known as the method of lower and upper solutions) is applied to computing a nonlinear difference scheme obtained after discretisation of the continuous problem. Secondly, a monotone domain decomposition algorithm based on a modification of the Schwarz alternating method is constructed. This monotone algorithm solves only linear discrete systems at each iterative step of the iterative process. The rate of convergence of the monotone Schwarz method is estimated. Numerical experiments are presented. [Copyright &y& Elsevier]