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The paper presents the numerical performance of the preconditioned generalized conjugate gradient (PGCG) methods in solving non‐linear convection — diffusion equations. Three non‐linear systems which describe a non‐isothermal chemical reactor, the chemically driven convection in a porous medium and the incompressible steady flow past a sphere are the test problems. The standard second order accurate centred finite difference scheme is used to discretize the models equations. The discrete approximations are solved with a double iterative process using the Newton method as outer iteration and the PGCG algorithm as inner iteration. Three PGCG techniques, which emerge to be the best performing, are tested. Laplace‐type operators are employed for preconditioning. The results show that the convergence of the PGCG methods depends strongly on the convection—diffusion ratio. The most robust algorithm is GMRES. But even with GMRES non‐convergence occurs when the convection—diffusion ratio exceeds a limit value. This value seems to be influenced by the non‐linearity type.

The paper analyses the preconditioning of non‐linear nonsymmetric equations with approximations of the discrete Laplace operator. The test problems are non‐linear 2‐D elliptic equations that describe natural convection, Darcy flow, in a porous medium. The standard second order accurate finite difference scheme is used to discretize the models’ equations. The discrete approximations are solved with a double iterative process using the Newton method as outer iteration and the preconditioned generalised conjugate gradient (PGCG) methods as inner iteration. Three PGCG algorithms, CGN, CGS and GMRES, are tested. The preconditioning with discrete Laplace operator approximations consists of replacing the solving of the equation with the preconditioner by a few iterations of an appropriate iterative scheme. Two iterative algorithms are tested: incomplete Cholesky (IC) and multigrid (MG). The numerical results show that MG preconditioning leads to mesh independence. CGS is the most robust algorithm but its efficiency is lower than that of GMRES.

A new class of approximate inverse banded matrix techniques based on the concept of LU‐type factorization procedures is introduced for computing explicitly approximate inverses without inverting the decomposition factors. Explicit preconditioned iterative schemes in conjunction with approximate inverse matrix techniques are presented for the efficient solution of banded linear systems. Applications of the method on a linear system are discussed and numerical results are given.

A computational procedure is presented for the efficient non‐linear dynamic analysis of quasi‐symmetric structures. The procedure is based on approximating the unsymmetric response vectors, at each time step, by a linear combination of symmetric and antisymmetric vectors, each obtained using approximately half the degrees of freedom of the original model. A mixed formulation is used with the fundamental unknowns consisting of the internal forces (stress resultants), generalized displacements and velocity components. The spatial discretization is done by using the finite element method, and the governing semi‐discrete finite element equations are cast in the form of first‐order non‐linear ordinary differential equations. The temporal integration is performed by using implicit multistep integration operators. The resulting non‐linear algebraic equations, at each time step, are solved by using iterative techniques. The three key elements of the proposed procedure are: (a) use of mixed finite element models with independent shape functions for the stress resultants, generalized displacements, and velocity components and with the stress resultants allowed to be discontinuous at interelement boundaries; (b) operator splitting, or restructuring of the governing discrete equations of the structure to delineate the contributions to the symmetric and antisymmetric vectors constituting the response; and (c) use of a two‐level iterative process (with nested iteration loops) to generate the symmetric and antisymmetric components of the response vectors at each time step. The top‐ and bottom‐level iterations (outer and inner iterative loops) are performed by using the Newton—Raphson and the preconditioned conjugate gradient (PCG) techniques, respectively. The effectiveness of the proposed strategy is demonstrated by means of a numerical example and the potential of the strategy for solving more complex non‐linear problems is discussed.

The purpose of this paper is to show benefits of deflated preconditioned conjugate gradients (CG) in the solution of transient, incompressible, viscous flows coupled with heat transfer.

This paper presents the implementation of deflated preconditioned CG as the iterative driver for the system of linearized equations for viscous, incompressible flows and heat transfer simulations. The De Sampaio-Coutinho particular form of the Petrov-Galerkin Generalized Least Squares finite element formulation is used in the discretization of the governing equations, leading to symmetric positive definite matrices, allowing the use of the CG solver.

The use of deflation techniques improves the spectral condition number. The authors show in a number of problems of coupled viscous flow and heat transfer that convergence is achieved with a lower number of iterations and smaller time.

This work addressed for the first time the use of deflated CG for the solution of transient analysis of free/forced convection in viscous flows coupled with heat transfer.

Two efficient computational procedures are presented for generating the global approximation vectors used in conjunction with the reduction methods for the large‐deflection non‐linear analysis of symmetric structures with unsymmetric boundary conditions. Both procedures are based on restructuring the governing equations for each of the unsymmetric global approximation vectors to delineate the different contributions to the symmetric and antisymmetric components of this vector. In the first procedure the unsymmetric global approximation vectors are approximated by linear combinations of symmetric and antisymmetric modes, which are generated by using the finite element method. The amplitudes of these modes are computed by using the classical Rayleigh‐Ritz technique. The second procedure is based on using a preconditioned conjugate gradient (PCG) technique for generating the global approximation vectors, and selecting the preconditioning matrix to be the matrix associated with the symmetric response. In both procedures the size of the analysis model used in generating the global approximation vectors is identical to that of the corresponding structure with symmetric boundary conditions. The similarities between the two procedures are identified, and their effectiveness is demonstrated by means of two numerical examples of large‐deflection, non‐linear static problems of shells.

To provide an analysis of transient heat conduction, which is solved using different iterative solvers for graduate and postgraduate students (researchers) which can help them develop their own research.

Three‐dimensional transient heat conduction in homogeneous materials using different time‐stepping methods such as finite difference (Θ explicit, implicit and Crank‐Nicolson) and finite element (weighted residual and least squared) methods. Iterative solvers used in the paper are conjugate gradient (CG), preconditioned gradient, least square CG, conjugate gradient squared (CGS), preconditioned CGS, bi‐conjugate gradient (BCG), preconditioned BCG, bi‐conjugate gradient stabilized (BCGSTAB), reconditioned BCGSTAB and Gaussian elimination with incomplete Cholesky factorization.

Provides information on which time‐stepping method is the most accurate, which solver is the fastest to solve a symmetric and positive system of linear matrix equations of all those considered.

Fortran 90 code given as an abstract can be very useful for graduate and postgraduate students to develop their own code.

This paper offers practical help to an individual starting his/her research in the finite element technique and numerical methods.

Error indicators are introduced as part of a simple computational procedure for improving the accuracy of the finite element solutions for plate and shell problems. The procedure is based on using an initial (coarse) grid and a refined (enriched) grid, and approximating the solution for the refined grid by a linear combination of a few global approximation vectors (or modes) which are generated by solving two uncoupled sets of equations in the coarse grid unknowns and the additional degrees of freedom of the refined grid. The global approximation vectors serve as error indicators since they provide quantitative pointwise information about the sensitivity of the different response quantities to the approximation used. The three key elements of the computational procedure are: (a) use of mixed finite element models with discontinuous stress resultants at the element interfaces; (b) operator splitting, or additive decomposition of the finite element arrays for the refined grid into the sum of the coarse grid arrays and correction terms (representing the refined grid contributions); and (c) application of a reduction method through successive use of the finite element method and the classical Bubnov—Galerkin technique. The finite element method is first used to generate a few global approximation vectors (or modes). Then the amplitudes of these modes are computed by using the Bubnov—Galerkin technique. The similarities between the proposed computational procedure and a preconditioned conjugate gradient (PCG) technique are identified and are exploited to generate from the PCG technique pointwise error indicators. The effectiveness of the proposed procedure is demonstrated by means of two numerical examples of an isotropic toroidal shell and a laminated anisotropic cylindrical panel.

The purpose of this paper is to propose novel parallel computational techniques for the parallelization of explicit finite element generalized approximate inverse methods, based on Portable Operating System Interface for UniX (POSIX) threads, for multicore systems.

The authors' main motive for the derivation of the new Parallel Generalized Approximate Inverse Finite Element Matrix algorithmic techniques is that they can be efficiently used in conjunction with explicit preconditioned conjugate gradient‐type schemes on multicore systems. The proposed parallelization technique of the Optimized Banded Generalized Approximate Inverse Finite Element Matrix (OBGAIFEM) algorithm is achieved based on the concept of the “fish bone” approach with the use of a thread pool pattern. Theoretical estimates on the computational complexity of the parallel generalized approximate inverse finite element matrix algorithmic techniques are also derived.

Application of the proposed method on a two‐dimensional boundary value problem is discussed and numerical results are given on a multicore system using POSIX threads. These results tend to become optimum and are favorably compared to corresponding results from multiprocessor systems, as presented in recent work by Gravvanis et al.

The proposed parallel explicit finite element generalized approximate inverse preconditioning, using approximate factorization and approximate inverse algorithms, is an efficient computational method that is valuable for computer scientists and for scientists and engineers in engineering computations.

Finite element discretizations of low‐frequency, time‐harmonic magnetic problems lead to sparse, complex symmetric systems of linear equations. The question arises which Krylov subspace methods are appropriate to solve such systems. The quasi minimal residual method combines a constant amount of work and storage per iteration step with a smooth convergence history. These advantages are obtained by building a quasi minimal residual approach on top of a Lanczos process to construct the search space. Solving the complex systems by transforming them to equivalent real ones of double dimension has to be avoided as such real systems have spectra that are less favourable for the convergence of Krylov‐based methods. Numerical experiments are performed on electromagnetic engineering problems to compare the quasi minimal residual method to the bi‐conjugate gradient method and the generalized minimal residual method.