Block s‐step Krylov iterative methods

Block (including s‐step) iterative methods for (non)symmetric linear systems have been studied and implemented in the past. In this article we present a (combined) block s‐step Krylov iterative method for nonsymmetric linear systems. We then consider the problem of applying any block iterative method to solve a linear system with one right‐hand side using many linearly independent initial residual vectors. We present a new algorithm which combines the many solutions obtained (by any block iterative method) into a single solution to the linear system. This approach of using block methods in order to increase the parallelism of Krylov methods is very useful in parallel systems. We implemented the new method on a parallel computer and we ran tests to validate the accuracy and the performance of the proposed methods. It is expected that the block s‐step methods performance will scale well on other parallel systems because of their efficient use of memory hierarchies and their reduction of the number of global communication operations over the standard methods. Copyright © 2009 John Wiley & Sons, Ltd.     

Two‐grid methods for time‐harmonic Maxwell equations

In this paper, we develop several two‐grid methods for the Nédélec edge finite element approximation of the time‐harmonic Maxwell equations. We first present a two‐grid method that uses a coarse space to solve the original problem and then use a fine space to solve a corresponding symmetric positive definite problem. Then, we present two types of iterative two‐grid methods, one is to add the kernel of the curl ‐operator in the fine space to a coarse mesh space to solve the original problem and the other is to use an inner iterative method for dealing with the kernel of the curl ‐operator in the fine space and the coarse space, separately. We provide the error estimates for the first two methods and present numerical experiments to show the efficiency of our methods.Copyright © 2012 John Wiley & Sons, Ltd.     

A Jacobi–Davidson method for two‐real‐parameter nonlinear eigenvalue problems arising from delay‐differential equations

The critical delays of a delay‐differential equation can be computed by solving a nonlinear two‐parameter eigenvalue problem. The solution of this two‐parameter problem can be translated to solving a quadratic eigenvalue problem of squared dimension. We present a structure preserving QR‐type method for solving such quadratic eigenvalue problem that only computes real‐valued critical delays; that is, complex critical delays, which have no physical meaning, are discarded. For large‐scale problems, we propose new correction equations for a Newton‐type or Jacobi–Davidson style method, which also forces real‐valued critical delays. We present three different equations: one real‐valued equation using a direct linear system solver, one complex valued equation using a direct linear system solver, and one Jacobi–Davidson style correction equation that is suitable for an iterative linear system solver. We show numerical examples for large‐scale problems arising from PDEs. Copyright © 2012 John Wiley & Sons, Ltd.     

On order‐reducible sinc discretizations and block‐diagonal preconditioning methods for linear third‐order ordinary differential equations

By introducing a variable substitution, we transform the two‐point boundary value problem of a third‐order ordinary differential equation into a system of two second‐order ordinary differential equations (ODEs). We discretize this order‐reduced system of ODEs by both sinc‐collocation and sinc‐Galerkin methods, and average these two discretized linear systems to obtain the target system of linear equations. We prove that the discrete solution resulting from the linear system converges exponentially to the true solution of the order‐reduced system of ODEs. The coefficient matrix of the linear system is of block two‐by‐two structure, and each of its blocks is a combination of Toeplitz and diagonal matrices. Because of its algebraic properties and matrix structures, the linear system can be effectively solved by Krylov subspace iteration methods such as GMRES preconditioned by block‐diagonal matrices. We demonstrate that the eigenvalues of certain approximation to the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the discretized linear system, and we use numerical examples to illustrate the feasibility and effectiveness of this new approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Mass‐lumping or not mass‐lumping for eigenvalue problems

In this article we analyze the effect of mass‐lumping in the linear triangular finite element approximation of second‐order elliptic eigenvalue problems. We prove that the eigenvalue obtained by using mass‐lumping is always below the one obtained with exact integration. For singular eigenfunctions, as those arising in non convex polygons, we prove that the eigenvalue obtained with mass‐lumping is above the exact eigenvalue when the mesh size is small enough. So, we conclude that the use of mass‐lumping is convenient in the singular case. When the eigenfunction is smooth several numerical experiments suggest that the eigenvalue computed with mass‐lumping is below the exact one if the mesh is not too coarse. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 653–664, 2003     

A new one‐shot pointwise source reconstruction method

In this work, a new pointwise source reconstruction method is proposed. From a single pair of boundary measurements, we want to completely characterize the unknown set of pointwise sources, namely, the number of sources and their locations and intensities. The idea is to rewrite the inverse source problem as an optimization problem, where a Kohn‐Vogelius type functional is minimized with respect to a set of admissible pointwise sources. The resulting second‐order reconstruction algorithm is non‐iterative and thus very robust with respect to noisy data. Finally, in order to show the effectiveness of the devised reconstruction algorithm, some numerical experiments into two spatial dimensions are presented. Copyright © 2016 John Wiley & Sons, Ltd.     

Interaction of faults under slip‐dependent friction. Non‐linear eigenvalue analysis

We analyse the evolution of a system of finite faults by considering the non‐linear eigenvalue problems associated to static and dynamic solutions on unbounded domains. We restrict our investigation to the first eigenvalue (Rayleigh quotient). We point out its physical significance through a stability analysis and we give an efficient numerical algorithm able to compute it together with the corresponding eigenfunction. We consider the anti‐plane shearing on a system of finite faults under a slip‐dependent friction in a linear elastic domain, not necessarily bounded. The static problem is formulated in terms of local minima of the energy functional. We introduce the non‐linear (static) eigenvalue problem and we prove the existence of a first eigenvalue/eigenfunction characterizing the isolated local minima. For the dynamic problem, we discuss the existence of solutions with an exponential growth, to deduce a (dynamic) non‐linear eigenvalue problem. We prove the existence of a first dynamic eigenvalue and we analyse its behaviour with respect to the friction parameter. We deduce a mixed finite element discretization of the non‐linear spectral problem and we give a numerical algorithm to approach the first eigenvalue/eigenfunction. Finally we give some numerical results which include convergence tests, on a single fault and a two‐faults system, and a comparison between the non‐linear spectral results and the time evolution results. Copyright © 2004 John Wiley & Sons, Ltd.     

Two‐level discretization of the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity

The r‐Laplacian has played an important role in the development of computationally efficient models for applications, such as numerical simulation of turbulent flows. In this article, we examine two‐level finite element approximation schemes applied to the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity, where r is the order of the power‐law artificial viscosity term. In the two‐level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single‐step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two‐level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We provide rigorous numerical analysis of the two‐level approximation scheme and derive scalings which vary based on the coefficient r, coarse mesh size H, fine mesh size h, and filter radius δ. We also investigate the two‐level algorithm in several computational settings, including the 3D numerical simulation of flow past a backward‐facing step at Reynolds number Re = 5100. In all numerical tests, the two‐level algorithm was proven to achieve the same order of accuracy as the standard one‐level algorithm, at a fraction of the computational cost. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

High‐order Padé and singly diagonally Runge‐Kutta schemes for linear ODEs,application to wave propagation problems

In this article, we address the problem of constructing high‐order implicit time schemes for wave equations. We consider two classes of one‐step A‐stable schemes adapted to linear Ordinary Differential Equation (ODE). The first class, which is not dissipative is based on the diagonal Padé approximant of exponential function. For this class, the obtained schemes have the same stability function as Gauss Runge‐Kutta (Gauss RK) schemes. They have the advantage to involve the solution of smaller linear systems at each time step compared to Gauss RK. The second class of schemes are constructed such that they require the inversion of a unique linear system several times at each time step like the Singly Diagonally Runge‐Kutta (SDIRK) schemes. While the first class of schemes is constructed for an arbitrary order of accuracy, the second‐class schemes is given up to order 12. The performance assessment we provide shows a very good level of accuracy for both classes of schemes, and the great interest of considering high‐order time schemes that are faster. The diagonal Padé schemes seem to be more accurate and more robust.     

A new two‐phase structure‐preserving doubling algorithm for critically singular M‐matrix algebraic Riccati equations

Among numerous iterative methods for solving the minimal nonnegative solution of an M‐matrix algebraic Riccati equation, the structure‐preserving doubling algorithm (SDA) stands out owing to its overall efficiency as well as accuracy. SDA is globally convergent and its convergence is quadratic, except for the critical case for which it converges linearly with the linear rate 1/2. In this paper, we first undertake a delineatory convergence analysis that reveals that the approximations by SDA can be decomposed into two components: the stable component that converges quadratically and the rank‐one component that converges linearly with the linear rate 1/2. Our analysis also shows that as soon as the stable component is fully converged, the rank‐one component can be accurately recovered. We then propose an efficient hybrid method, called the two‐phase SDA, for which the SDA iteration is stopped as soon as it is determined that the stable component is fully converged. Therefore, this two‐phase SDA saves those SDA iterative steps that previously have to have for the rank‐one component to be computed accurately, and thus essentially, it can be regarded as a quadratically convergent method. Numerical results confirm our analysis and demonstrate the efficiency of the new two‐phase SDA. Copyright © 2015 John Wiley & Sons, Ltd.     

On the subdomain‐Galerkin/least squares method for 2‐ and 3‐D mixed elliptic problems with reaction terms

In this article we apply the subdomain‐Galerkin/least squares method, which is first proposed by Chang and Gunzburger for first‐order elliptic systems without reaction terms in the plane, to solve second‐order non‐selfadjoint elliptic problems in two‐ and three‐dimensional bounded domains with triangular or tetrahedral regular triangulations. This method can be viewed as a combination of a direct cell vertex finite volume discretization step and an algebraic least‐squares minimization step in which the pressure is approximated by piecewise linear elements and the flux by the lowest order Raviart‐Thomas space. This combined approach has the advantages of both finite volume and least‐squares methods. Among other things, the combined method is not subject to the Ladyzhenskaya‐Babus?ka‐Brezzi condition, and the resulting linear system is symmetric and positive definite. An optimal error estimate in the H¹(Ω) × H(div; Ω) norm is derived. An equivalent residual‐type a posteriori error estimator is also given. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 738–751, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10030.     

An iterative method based on equation decomposition for the fourth‐order singular perturbation problem

In this article, we propose an iterative method based on the equation decomposition technique ( 1 ) for the numerical solution of a singular perturbation problem of fourth‐order elliptic equation. At each step of the given method, we only need to solve a boundary value problem of second‐order elliptic equation and a second‐order singular perturbation problem. We prove that our approximate solution converges to the exact solution when the domain is a disc. Our numerical examples show the efficiency and accuracy of our method. Our iterative method works very well for singular perturbation problems, that is, the case of 0 < ε ? 1, and the convergence rate is very fast. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

On the Odir iterative method for non‐symmetric indefinite linear systems

Several Krylov subspace iterative methods have been proposed for the approximation of the solution of general non‐symmetric linear systems. Odir is such a method. Here we study the restarted version of Odir for non‐symmetric indefinite linear systems and we prove convergence under certain conditions on the matrix of coefficients. These results hold for all the restarted Krylov methods equivalent to Odir. We also introduce a new truncated Odir method which is proved to converge for a large class of non‐symmetric indefinite linear systems. This new method requires one‐half of the storage of the standard Odir. Copyright © 2001 John Wiley & Sons, Ltd.     

Non‐elementary speed‐ups in logic calculi

In this paper we show some non‐elementary speed‐ups in logic calculi: Both a predicative second‐order logic and a logic for fixed points of positive formulas are shown to have non‐elementary speed‐ups over first‐order logic. Also it is shown that eliminating second‐order cut formulas in second‐order logic has to increase sizes of proofs super‐exponentially, and the same in eliminating second‐order epsilon axioms. These are proved by relying on results due to P. Pudlák. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal uniform error estimates for moving least‐squares collocation with application to option pricing under jump‐diffusion processes

In this study, we derive optimal uniform error bounds for moving least‐squares (MLS) mesh‐free point collocation (also called finite point method) when applied to solve second‐order elliptic partial integro‐differential equations (PIDEs). In the special case of elliptic partial differential equations (PDEs), we show that our estimate improves the results of Cheng and Cheng (Appl. Numer. Math. 58 (2008), no. 6, 884–898) both in terms of the used error norm (here the uniform norm and there the discrete vector norm) and the obtained order of convergence. We then present optimal convergence rate estimates for second‐order elliptic PIDEs. We proceed by some numerical experiments dealing with elliptic PDEs that confirm the obtained theoretical results. The article concludes with numerical approximation of the linear parabolic PIDE arising from European option pricing problem under Merton's and Kou's jump‐diffusion models. The presented computational results (including the computation of option Greeks) and comparisons with other competing approaches suggest that the MLS collocation scheme is an efficient and reliable numerical method to solve elliptic and parabolic PIDEs arising from applied areas such as financial engineering.     

On a boundary variational inequality of the second kind modelling a friction problem

A scalar contact problem with friction governed by the Yukawa equation is reduced to a boundary variational inequality. The presence of the non‐differentiable friction functional causes some difficulties when approximated. We present two methods to overcome this difficulty. The first one is a regularization leading to a non‐linear boundary variational equation, for which we propose an iterative procedure, whereas the second method is based on the boundary mixed variational formulation involving Lagrange multipliers. We propose Uzawa's algorithm to compute the saddle point of the corresponding boundary Lagrangian and investigate the discretization of various formulations by the boundary element Galerkin method. Convergence of the boundary element solution is proved and a convergence order is obtained. Copyright © 2002 John Wiley & Sons, Ltd.     

A FEM–DtN formulation for a non‐linear exterior problem in incompressible elasticity

In this paper, we combine the usual finite element method with a Dirichlet‐to‐Neumann (DtN) mapping, derived in terms of an infinite Fourier series, to study the solvability and Galerkin approximations of an exterior transmission problem arising in non‐linear incompressible 2d‐elasticity. We show that the variational formulation can be written in a Stokes‐type mixed form with a linear constraint and a non‐linear main operator. Then, we provide the uniqueness of solution for the continuous and discrete formulations, and derive a Cea‐type estimate for the associated error. In particular, our error analysis considers the practical case in which the DtN mapping is approximated by the corresponding finite Fourier series. Finally, a reliable a posteriori error estimate, well suited for adaptive computations, is also given. Copyright © 2003 John Wiley & Sons, Ltd.     

Newton‐Krylov iterative matrix representative spectrum

The concept of a representative spectrum is introduced in the context of Newton‐Krylov methods. This concept allows a better understanding of convergence rate accelerating techniques for Krylov‐subspace iterative methods in the context of CFD applications of the Newton‐Krylov approach to iteratively solve sets of non‐linear equations. The dependence of the representative spectrum on several parameters such as mesh, Mach number or discretization techniques is studied and analyzed. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

On a Chebyshev accelerated splitting iteration method with application to two‐by‐two block linear systems

The Chebyshev accelerated preconditioned modified Hermitian and skew‐Hermitian splitting (CAPMHSS) iteration method is presented for solving the linear systems of equations, which have two‐by‐two block coefficient matrices. We derive an iteration error bound to show that the new method is convergent as long as the eigenvalue bounds are not underestimated. Even when the spectral information is lacking, the CAPMHSS iteration method could be considered as an exponentially converging iterative scheme for certain choices of the method parameters. In this case, the convergence rate is independent of the parameters. Besides, the linear subsystems in each iteration can be solved inexactly, which leads to the inexact CAPMHSS iteration method. The iteration error bound of the inexact method is derived also. We discuss in detail the implementation of CAPMHSS for solving two models arising from the Galerkin finite‐element discretizations of distributed control problems and complex symmetric linear systems. The numerical results show the robustness and the efficiency of the new methods.     

A coarse–fine‐mesh stabilization for an alternating Schwarz domain decomposition method

Domain decomposition methods can be solved in various ways. In this paper, domain decomposition in strips is used. It is demonstrated that a special version of the Schwarz alternating iteration method coupled with coarse–fine‐mesh stabilization leads to a very efficient solver, which is easy to implement and has a behavior nearly independent of mesh and problem parameters. The novelty of the method is the use of alternating iterations between odd‐ and even‐numbered subdomains and the replacement of the commonly used coarse‐mesh stabilization method with coarse–fine‐mesh stabilization.