SUMMARYThe focus of this paper is on the numerical solution of large sparse non-linear systems of algebraic equations on parallel computers. Such non-linear systems often arise from the discretization of non-linear partial di erential equations, such as the Navier-Stokes equations for uid ows, using ÿnite element or ÿnite di erence methods. A traditional inexact Newton method, applied directly to the discretized system, does not work well when the non-linearities in the algebraic system become unbalanced. In this paper, we study some preconditioned inexact Newton algorithms, including the single-level and multilevel non-linear additive Schwarz preconditioners. Some results for solving the high Reynolds number incompressible Navier-Stokes equations are reported.
In this paper, simulations of I-V characteristics and band structures of magnesium and silicon doped gallium nitride diodes are presented. The numerical algorithm is based on the drift-diffusion semi-classic model, with the van Roosbroeck differential equation system involved. The model accounts for trap-assisted tunneling, which provides better agreement between the predicted and experimental I-V characteristics of p-n junctions in the low-bias range. We have performed one-dimensional simulations of devices. We compare the results with the results obtained from the standard drift-diffusion model. It is shown that taking the trap-assisted tunneling into account leads to good agreement with experimental data. We also demonstrate that a high doping of the p-n junctions may significantly increase the nonradiative recombination rate due to the prescribed effect.
We propose two variants of the overlapping additive Schwarz method for the finite element discretization of the elliptic problem in 3D with highly heterogeneous coefficients. The methods are efficient and simple to construct using the abstract framework of the additive Schwarz method, and an idea of adaptive coarse spaces. In one variant, the coarse space consists of finite element functions associated with the wire basket nodes and functions based on solving some generalized eigenvalue problem on the faces, and in the other variant, it contains functions associated with the vertex nodes with functions based on solving some generalized eigenvalue problems on subdomain faces and on subdomain edges. The functions that are used to build the coarse spaces are chosen adaptively, they correspond to the eigenvalues that are smaller than a given threshold. The convergence rate of the preconditioned conjugate gradients method in both cases, is shown to be independent of the variations in the coefficients for sufficient number of eigenfunctions in the coarse space. Numerical results are given to support the theory.being the layer of all elements along the subdomain boundary ∂Ω i , α the varying coefficient, and C a constant. Because the α is varying, estimating the L 2 term say using the Poincaré or a weighted Poincaré inequality will necessarily introduce the contrast into the estimate, i.e. the ratio between the largest and the smallest values
An additive Schwarz preconditioner for nonconforming mortar finite element discretization of a second order elliptic problem in two dimensions with arbitrary large jumps of the discontinuous coefficients in subdomains is described.An almost optimal estimate of the condition number of the preconditioned problem is proved. The number of preconditioned conjugate gradient iterations is independent of jumps of the coefficients and is proportional to (1 + log(H/h)), where H, h are mesh sizes. (2000): 65N55, 65N30, 65N22.
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