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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]

Nonoverlapping domain decomposition methods have been shown to provide efficient iterative algorithms for the solution of the time-harmonic Maxwell's equations. Convergence of the algorithms depends strongly upon the nature of the transmission conditions that communicate information between adjacent subdomains. In this work, we introduce a new second order transmission condition that improves convergence. In contrast to previous high order interface conditions, the new condition uses two second order transverse derivatives to improve the convergence of evanescent modes without deteriorating the convergence of propagating modes. An analysis using a splitting of the field into traverse electric and magnetic components demonstrates the improved convergence provided by the transmission condition. Numerical experiments demonstrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, optimized Schwarz methods with Robin transmission conditions are considered for solving the time-dependent linear algebraic systems resulting from finite element approximations for parabolic problems. We use both analysis and numerical experiments to investigate the convergence behavior. Particularly, we check the influence of the time step size on the convergence rate. We get the estimate of the upper bound of convergence rate 1 - O(min{h1/2H-1/2, h1/2H3/2t-1}), where h is the mesh size, H is the size of subdomains, and t is the time step size. Our numerical results show that the convergence rate of this method is fast. We also search for the optimal parameter λ by experiments and find the rule of its distribution. [ABSTRACT FROM AUTHOR]

In recent years, a nonoverlapping Robin-type domain decomposition method (DDM) for the finite element discretization systems of the second order elliptic equations, which is based on using Robin-type boundary conditions as information transmission conditions on the subdomain interfaces, has been developed and analyzed since it was first proposed by P. L. Lions in [On the Schwarz alternating method III: A variant for nonoverlapping subdomains, in Proceedings of the 3rd International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, PA, 1990, pp. 202-223]. However, the convergence rate of this DDM with many subdomains remains open when the lower term of equations vanishes. This open problem will be considered in this paper. The convergence rate is almost 1 — O(h¹/²H-1/²) in certain cases—for example, the case of a small number of subdomains, where h is the mesh size and H is the size of subdomain. In order to get the desirous convergence results, two mathematics skills are introduced in this paper; one is complexification of real linear space and the other is the spectral radius formula. [ABSTRACT FROM AUTHOR]

A class of corrected explicit-implicit domain decomposition (CEIDD) methods is investigated for the parallel approximation of linear heat equations. Explicit-implicit domain decomposition (EIDD) methods are computationally and communicationally efficient for each time step but always suffer from small time step size restrictions. By adding an interface correction step to Kuznetsov's EIDD, the one-dimensional CEIDD procedure achieves unconditional stability without discarding the time-stepwise efficiency of the EIDD method. In order to maintain the virtues of the CEIDD method and improve the flexibility in domain partitioning, for solving multidimensional problems, special zigzag-shaped interfaces are suggested in the CEIDD method. Based on non-crossover and crossover types of zigzag interfaces, the resulting CEIDD-ZI algorithms are studied for two strategies of subdomain partition. By the energy method, it shows that the proposed algorithms, including their degenerate cases—the corrected explicit hopscotch schemes-are convergent in the discrete H¹ seminorm and L² norm. Numerical experiments confirm the results in our analysis. [ABSTRACT FROM AUTHOR]

In this paper, we propose a nonoverlapping domain decomposition method for solving the three-dimensional Maxwell equations, based on the edge element discretization. For the Schur complement system on the interface, we construct an efficient preconditioner by introducing two special coarse subspaces defined on the nonoverlapping sub domains. It is shown that the condition number of the preconditioned system grows only polylogarithmically with the ratio between the subdomain diameter and the finite element mesh size but possibly depends on the jumps of the coefficients. [ABSTRACT FROM AUTHOR]