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In this paper, we present a two-level additive Schwarz method for solving a system arising from the discretization of the nonselfadjoint elliptic equation. By employing the Cauchy-Schwarz-type inequality and stable decomposition under the energy norm, we obtain the optimal convergence theory for the proposed method. It shows that the convergence rate is bounded and independent of the fine mesh size and the number of subdomains. Some numerical results are reported to verify our theoretical result. Moreover, we demonstrate the benefit compared to the classical two-level additive Schwarz algorithm for solving convection-diffusion equations.

In this paper, a new multiplicative Schwarz method is presented for solving a system arising from the discretization of the nonselfadjoint elliptic equations. In the implementation, we apply the pr...

In this paper, we present a two-level space–time hybrid Schwarz preconditioner for GMRES based on the classical hybrid Schwarz method and the second-order backward differentiation formula. In the proposed method, the parabolic equations are solved in parallel on both of the space and time directions. Under some reasonable assumptions, the optimal convergence theory is developed for the proposed space–time method, i.e., the convergence rate is independent of the mesh parameters, the number of subdomains and the window size. Some numerical results are given to confirm the theory very well in terms of the convergence rate and accuracy. And the strong/weak scalability obtained with 4096 processors is also reported to show the efficiency of the proposed method.

In the past few years, the number of processor cores of top ranked supercomputers has increased drastically. It is challenging to design efficient parallel algorithms that offer such a high degree of parallelism, especially for certain time-dependent problems because of the sequential nature of “time”. To increase the degree of parallelization, some parallel-in-time algorithms have been developed. In this paper, we give an overview of some recently introduced parallel-in-time methods, and present in detail the class of space-time Schwarz methods, including the standard and the restricted versions, for solving parabolic partial differential equations. Some numerical experiments carried out on a parallel computer with a large number of processor cores for three-dimensional problems are given to show the parallel scalability of the methods. In the end of the paper, we provide a comparison of the parallel-in-time algorithms with a traditional algorithm that is parallelized only in space.

In this paper, we give a second order in time incremental pressure correction finite element method for the Navier-Stokes/Darcy problem. In this method, the Navier-Stokes/Darcy problem is solved in three steps: a convection-diffusion step, a projection correction (incremental pressure correction) step and a Darcy step. In this way, the Navier-Stokes/Darcy equation is solved in a fractional step way, which is a decoupled method. In order to decouple the equation, we use the numerical solutions at the last time level to give the interface conditions. The stability analysis shows that the second order in time incremental pressure correction finite element method is unconditionally stable. The optimal error estimate is also given. Finally, we present some numerical results to show the efficiency of the method.

In this paper, we consider the posteriori error estimates of Galerkin finite element method for the unsteady Navier–Stokes equations. By constructing the approximate Navier–Stokes reconstructions, the errors of velocity and pressure are split into two parts. For the estimates of time part, the energy method and other skills are used, for the estimates of spatial part, the well-developed theoretical analysis of posteriori error estimates for the elliptic problem can be adopted. More important, the error estimates of time part can be controlled by the estimates of spatial part. As a consequence, the posteriori error estimates in L∞(0, T; L2(Ω)), L∞(0, T; H1(Ω)) and L2(0, T; L2(Ω)) norms for velocity and pressure are derived in both spatial discrete and time-space fully discrete schemes.

We present and investigate two new robust space–time preconditioned GMRES methods for solving linear system of equations arising from the discretization of parabolic equations. With these methods, a block diagonal linear system instead of a block lower bidiagonal linear system will be solved in each space–time subdomain. It shows that this technique can reduce the cost of the computation and memory overhead. We develop an optimal convergence theory which shows that convergence rate is bounded independently of the spatial mesh sizes, the time step size, the number of subdomains, the number of levels and the window size. Some numerical results are reported to confirm the theory in terms of optimality and scalability. Moreover, numerical comparisons with space–time additive Schwarz algorithm are also given to show the actual competitiveness our proposed space–time methods.

In this paper, a local multilevel algorithm is investigated for solving linear systems arising from adaptive finite element approximations of second order elliptic problems with smooth complex coefficients. It is shown that the abstract theory for local multilevel algorithm can also be applied to elliptic problems whose dominant coefficient is complex valued. Assuming that the coarsest mesh size is sufficiently small, we prove that this algorithm with Gauss-Seidel smoother is convergent and optimal on the adaptively refined meshes generated by the newest vertex bisection algorithm. Numerical experiments are reported to confirm the theoretical analysis.

In this paper, we investigate a class of multigrid methods with HSS smoothers applied to second order elliptic problems whose dominant coefficient is complex valued. These smoothers are based on Hermitian/skew-Hermitian splitting (HSS), modified HSS (MHSS) and preconditioned MHSS (PMHSS) iterations. It is shown that the abstract theory for multigrid method can be extended and applied to elliptic problems with complex coefficient. Under some assumption conditions, we prove that multigrid method with PMHSS smoother is convergent and optimal. Numerical results are reported to confirm the theoretical analysis.

The additive Schwarz (AS) method was originally designed for preconditioning elliptic problems, and recently the method was extended successfully as a coupled space-time preconditioner for parabolic problems. However, the existing theory for the additive Schwarz method doesn't apply directly to the space-time discretized problems. In this paper, we develop an optimal convergence theory for the two-level space-time additive Schwarz preconditioner. We obtain lower and upper bounds for the spectrum of the AS preconditioned operator and their dependency on the fine/coarse mesh sizes in space and time, the number of subdomains, and the window size (i.e., the number of coupled time steps). Some numerical experiments are reported to confirm the theory.

Precipitation behavior of AlMgCu alloy during S phase aging sequence was studied with high resolution transmission electron microscope (HRTEM) and selected area diffraction (SAED) to determine the existence of S″ phase and GPBII zone in the alloy. Fast Fourier transform (FFT) patterns corresponding to GPB zones at different aging stages were observed, and it is revealed that S phase forms instantly after the GPB zone dissolves during aging. Both simulated HRTEM and SAED images illustrate that the S″ phase shows exactly the same characteristics as the S phase does, which is consistent with the experimental observations. A unique phase was also observed, which produced extra diffraction dots near the {012}Al positions in FFT pattern. This phase was once considered to be either S″ phase or GPBII zone, but none of them has been affirmed so far. In this work, it is found that the unique phase is a distorted and rotated S phase and that S″ phase or GPBII zone does not form during the aging. We further show that the S precipitation sequence of the aged AlMgCu alloy should be SSS → Cu–Mg clusters/GPB zone → S′/S(Al2CuMg).

In this paper, a modified characteristics finite element method for the time dependent Navier–Stokes/Darcy problem with the Beavers–Joseph–Saffman interface condition is presented. In this method, the Navier–Stokes/Darcy equation is decoupled into two equations, one is the Navier–Stokes equation, the other is the Darcy equation, and the Navier–Stokes equation is solved by the modified characteristics finite element method. The theory analysis shows that this method has a good convergence property. In order to show the effect of our method, some numerical results was presented. The numerical results show that this method is highly efficient. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, the multigrid methods using Hermitian/skew-Hermitian splitting (HSS) iteration as smoothers are investigated. These smoothers also include the modified additive and multiplicative smoothers which result from subspace decomposition. Without full elliptic regularity assumption, it is shown that the multigrid methods with these smoothers converge uniformly for second-order nonselfadjoint elliptic boundary value problems if the mesh size of the coarsest grid is sufficiently small (but independent of the number of the multigrid levels). Numerical results are reported to confirm the theoretical analysis.

Effects of Ce addition on microstructure of Al 20 Cu 2 Mn 3 twin phase and mechanical properties of an Al–Cu–Mn casting alloy were investigated by transmission electron microscopy, selected area electron diffraction, high resolution transmission electron microscopy and tensile test. The results show that rare earth element Ce can retard the formation of the Al 20 Cu 2 Mn 3 phase in the Al–Cu–Mn alloy. Compared with the Ce containing alloy, patterns of particles of the Al 20 Cu 2 Mn 3 phase in the Al–Cu–Mn free Ce alloy are more diverse. The symmetry of neighboring components of twins is characterized by glide reflection and reflection. In addition, twins of the Al 20 Cu 2 Mn 3 phase can enhance the mechanical properties of the Al–Cu–Mn alloy.

In this paper, several two-grid algorithms are presented. For nonsymmetric linear systems, we propose a two-grid algorithm by using the information of the adjoint operator. The solution of the original systems is mainly reduced to a solution of symmetric positive definite (SPD) systems. For nonlinear systems, we present a two-grid algorithm based on the modified Newton method. The solution of the original systems on the fine space is reduced to the solution of two small systems on the coarse space and two similar linear systems (with same stiffness matrix) on the fine space. It is shown that the accuracy (L2 norm) obtained by this algorithm is as same as the optimal accuracy derived by using two full Newton steps. Additionally, for more practically applications, the ideas of these algorithms can be also extended to the multilevel case. Numerical experiments are given for these new algorithms.

Conjugate gradient methods are typically used to solve large scale unconstrained optimization problems. Recently, Hager and Zhang (2006) proposed two guaranteed descent conjugate gradient methods. In this paper, following Hager and Zhang (2006), we will use the modified secant condition given by Zhang et al.(1999) to present two new descent conjugate gradient methods. An interesting feature of these new methods is that they take both the gradient and function value information. Under some suitable assumptions, global convergence properties for these methods are established. Numerical comparisons with the Hager-Zhang methods are given.