Authors:Huiru Wang; Chengjian Zhang Abstract: This paper deals with the numerical methods for solving singular initial value problems. By adapting the block boundary value methods (BBVMs) for regular initial value problems, a class of adapted BBVMs are constructed for singular initial value problems. It is proved under some suitable conditions that the adapted BBVMs are uniquely solvable, stable and convergent of order p, where p is the consistence order of the methods. Several numerical examples are performed to verify the stability, efficiency and accuracy of the adapted methods. Moreover, a comparison between the adapted BBVMs and the IEM-based iterated defect correction methods is given. The numerical results show that the adapted BBVMs are comparable. PubDate: 2018-05-17 DOI: 10.1007/s10092-018-0264-5 Issue No:Vol. 55, No. 2 (2018)

Authors:Gabriel N. Gatica; Mauricio Munar; Filánder A. Sequeira Abstract: In this work we introduce and analyze a mixed virtual element method for the two-dimensional nonlinear Brinkman model of porous media flow with non-homogeneous Dirichlet boundary conditions. For the continuous formulation we consider a dual-mixed approach in which the main unknowns are given by the gradient of the velocity and the pseudostress, whereas the velocity itself and the pressure are computed via simple postprocessing formulae. In addition, because of analysis reasons we add a redundant term arising from the constitutive equation relating the pseudostress and the velocity, so that the well-posedness of the resulting augmented formulation is established by using known results from nonlinear functional analysis. Then, we introduce the main features of the mixed virtual element method, which employs an explicit piecewise polynomial subspace and a virtual element subspace for approximating the aforementioned main unknowns, respectively. In turn, the associated computable discrete nonlinear operator is defined in terms of the \(\mathbb {L}^2\) -orthogonal projector onto a suitable space of polynomials, which allows the explicit integration of the terms involving deviatoric tensors that appear in the original setting. Next, we show the well-posedness of the discrete scheme and derive the associated a priori error estimates for the virtual element solution as well as for the fully computable projection of it. Furthermore, we also introduce a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of this unknown with respect to the broken \(\mathbb {H}(\mathbf {div})\) -norm. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented. PubDate: 2018-05-17 DOI: 10.1007/s10092-018-0262-7 Issue No:Vol. 55, No. 2 (2018)

Authors:R. Verfürth Abstract: Motivated by stochastic convection–diffusion problems we derive a posteriori error estimates for non-stationary non-linear convection–diffusion equations acting as a deterministic paradigm. The problem considered here neither fits into the standard linear framework due to its non-linearity nor into the standard non-linear framework due to the lacking differentiability of the non-linearity. Particular attention is paid to the interplay of the various parameters controlling the relative sizes of diffusion, convection, reaction and non-linearity (noise). PubDate: 2018-05-08 DOI: 10.1007/s10092-018-0263-6 Issue No:Vol. 55, No. 2 (2018)

Authors:E. Bänsch; F. Karakatsani; C. G. Makridakis Abstract: This work is devoted to a posteriori error analysis of fully discrete finite element approximations to the time dependent Stokes system. The space discretization is based on popular stable spaces, including Crouzeix–Raviart and Taylor–Hood finite element methods. Implicit Euler is applied for the time discretization. The finite element spaces are allowed to change with time steps and the projection steps include alternatives that is hoped to cope with possible numerical artifices and the loss of the discrete incompressibility of the schemes. The final estimates are of optimal order in \(L^\infty (L^2) \) for the velocity error. PubDate: 2018-05-02 DOI: 10.1007/s10092-018-0259-2 Issue No:Vol. 55, No. 2 (2018)

Authors:Stephen Edward Moore Abstract: This paper is concerned with the analysis of a new stable space–time finite element method (FEM) for the numerical solution of parabolic evolution problems in moving spatial computational domains. The discrete bilinear form is elliptic on the FEM space with respect to a discrete energy norm. This property together with a corresponding boundedness property, consistency and approximation results for the FEM spaces yield an a priori discretization error estimate with respect to the discrete norm. Finally, we confirm the theoretical results with numerical experiments in spatial moving domains. PubDate: 2018-04-16 DOI: 10.1007/s10092-018-0261-8 Issue No:Vol. 55, No. 2 (2018)

Authors:Paola Brianzi; Fabio Di Benedetto; Claudio Estatico; Luca Surace Abstract: When iterative methods are employed as regularizers of inverse problems, a main issue is the trade-off between smoothing effects and computation time, related to the convergence rate of iterations. Very often, faster methods obtain less accuracy. A new acceleration strategy is presented here, inspired by a choice of penalty terms formerly proposed in 2012 by Huckle and Sedlacek in the context of Tikhonov regularization by direct solvers. More precisely, we consider a special penalty term endowed with high regularization capabilities, and we apply it by using the opposite sign, that is negative, to its regularization parameter. This unprecedented choice leads to an “irregularization” phenomenon, which speeds up the underlying basic iterative method. The speeding up effects of the negative valued penalty term can be controlled through a sequence of decreasing coefficients as the iterations proceed in order to prevent noise amplification, tuning the weight of the correction term which generates the anti-regularization behavior. Filter factor expansion and convergence are analyzed in the simplified context of linear inverse problems in Hilbert spaces, by considering modified Landweber iterations as a first case study. PubDate: 2018-04-10 DOI: 10.1007/s10092-018-0260-9 Issue No:Vol. 55, No. 2 (2018)

Authors:J. K. Liu; Y. M. Feng; L. M. Zou Abstract: The three-term conjugate gradient methods solving large-scale optimization problems are favored by many researchers because of their nice descent and convergent properties. In this paper, we extend some new conjugate gradient methods, and construct some three-term conjugate gradient methods. An remarkable property of the proposed methods is that the search direction always satisfies the sufficient descent condition without any line search. Under the standard Wolfe line search, the global convergence properties of the proposed methods are proved merely by assuming that the objective function is Lipschitz continuous. Preliminary numerical results and comparisons show that the proposed methods are efficient and promising. PubDate: 2018-03-16 DOI: 10.1007/s10092-018-0258-3 Issue No:Vol. 55, No. 2 (2018)

Authors:Falguni Roy; D. K. Gupta; Predrag S. Stanimirović Abstract: An interval extension of successive matrix squaring (SMS) method for computing the weighted Moore–Penrose inverse \(A^{\dagger }_{MN}\) along with its rigorous error bounds is proposed for given full rank \(m \times n\) complex matrices A, where M and N be two Hermitian positive definite matrices of orders m and n, respectively. Starting with a suitably chosen complex interval matrix containing \(A^{\dagger }_{MN}\) , this method generates a sequence of complex interval matrices each enclosing \(A^{\dagger }_{MN}\) and converging to it. A new method is developed for constructing initial complex interval matrix containing \(A^{\dagger }_{MN}\) . Convergence theorems are established. The R-order convergence is shown to be equal to at least l, where \(l \ge 2\) . A number of numerical examples are worked out to demonstrate its efficiency and effectiveness. Graphs are plotted to show variations of the number of iterations and computational times compared to matrix dimensions. It is observed that ISMS is more stable compared to SMS. PubDate: 2018-03-15 DOI: 10.1007/s10092-018-0257-4 Issue No:Vol. 55, No. 2 (2018)

Authors:M. H. Noori Skandari; M. Ghaznavi Abstract: Chebyshev pseudo-spectral method is one of the most efficient methods for solving continuous-time optimization problems. In this paper, we utilize this method to solve the general form of shortest path problem. Here, the main problem is converted into a nonlinear programming problem and by solving of which, we obtain an approximate shortest path. The feasibility of the nonlinear programming problem and the convergence of the method are given. Finally, some numerical examples are considered to show the efficiency of the presented method over the other methods. PubDate: 2018-02-23 DOI: 10.1007/s10092-018-0256-5 Issue No:Vol. 55, No. 1 (2018)

Authors:H. Barkouki; A. H. Bentbib; M. Heyouni; K. Jbilou Abstract: In this paper, we propose an extended block Krylov process to construct two biorthogonal bases for the extended Krylov subspaces \(\mathbb {K}_{m}^e(A,V)\) and \(\mathbb {K}_{m}^e(A^{T},W)\) , where \(A \in \mathbb {R}^{n \times n}\) and \(V,~W \in \mathbb {R}^{n \times p}\) . After deriving some new theoretical results and algebraic properties, we apply the proposed algorithm with moment matching techniques for model reduction in large scale dynamical systems. Numerical experiments for large and sparse problems are given to show the efficiency of the proposed method. PubDate: 2018-02-21 DOI: 10.1007/s10092-018-0248-5 Issue No:Vol. 55, No. 1 (2018)

Authors:Vladislav V. Kravchenko; Sergii M. Torba Abstract: A Neumann series of Bessel functions (NSBF) representation for solutions of Sturm–Liouville equations and for their derivatives is obtained. The representation possesses an attractive feature for applications: for all real values of the spectral parameter \(\omega \) the estimate of the difference between the exact solution and the approximate one (the truncated NSBF) depends on N (the truncation parameter) and the coefficients of the equation and does not depend on \(\omega \) . A similar result is valid when \(\omega \in {\mathbb {C}}\) belongs to a strip \(\left \hbox {Im }\omega \right <C\) . This feature makes the NSBF representation especially useful for applications requiring computation of solutions for large intervals of \(\omega \) . Error and decay rate estimates are obtained. An algorithm for solving initial value, boundary value or spectral problems for the Sturm–Liouville equation is developed and illustrated on a test problem. PubDate: 2018-02-19 DOI: 10.1007/s10092-018-0254-7 Issue No:Vol. 55, No. 1 (2018)

Authors:Bei Zhang; Jikun Zhao; Shaochun Chen; Yongqin Yang Abstract: In this paper, we propose a locking-free stabilized mixed finite element method for the linear elasticity problem, which employs a jump penalty term for the displacement approximation. The continuous piecewise k-order polynomial space is used for the stress and the discontinuous piecewise \((k-1)\) -order polynomial space for the displacement, where we require that \(k\ge 3\) in the two dimensions and \(k\ge 4\) in the three dimensions. The method is proved to be stable and k-order convergent for the stress in \(H(\mathrm {div})\) -norm and for the displacement in \(L^2\) -norm. Further, the convergence does not deteriorate in the nearly incompressible or incompressible case. Finally, the numerical results are presented to illustrate the optimal convergence of the stabilized mixed method. PubDate: 2018-02-19 DOI: 10.1007/s10092-018-0255-6 Issue No:Vol. 55, No. 1 (2018)

Authors:Konstantinos Spiliotis; Lucia Russo; Francesco Giannino; Salvatore Cuomo; Constantinos Siettos; Gerardo Toraldo Abstract: We address and discuss the application of nonlinear Galerkin methods for the model reduction and numerical solution of partial differential equations (PDE) with Turing instabilities in comparison with standard (linear) Galerkin methods. The model considered is a system of PDEs modelling the pattern formation in vegetation dynamics. In particular, by constructing the approximate inertial manifold on the basis of the spectral decomposition of the solution, we implement the so-called Euler–Galerkin method and we compare its efficiency and accuracy versus the linear Galerkin methods. We compare the efficiency of the methods by (a) the accuracy of the computed bifurcation points, and, (b) by the computation of the Hausdorff distance between the limit sets obtained by the Galerkin methods and the ones obtained with a reference finite difference scheme. The efficiency with respect to the required CPU time is also accessed. For our illustrations we used three different ODE time integrators, from the Matlab ODE suite. Our results indicate that the performance of the Euler–Galerkin method is superior compared to the linear Galerkin method when either explicit or linearly implicit time integration scheme are adopted. For the particular problem considered, we found that the dimension of approximate inertial manifold is strongly affected by the lenght of the spatial domain. Indeeed, we show that the number of modes required to accurately describe the long time Turing pattern forming solutions increases as the domain increases. PubDate: 2018-02-12 DOI: 10.1007/s10092-018-0245-8 Issue No:Vol. 55, No. 1 (2018)

Authors:Alexandru I. Mitrea Abstract: This paper deals with interpolatory product integration rules based on Jacobi nodes, associated with the Banach space of all s-times continuously differentiable functions, and with a Banach space of absolutely integrable functions, on the interval \([-1,1]\) of the real axis. In order to highlight the topological structure of the set of unbounded divergence for the corresponding product quadrature formulas, a family of continuous linear operators associated with these product integration procedures is pointed out, and the unboundedness of the set of their norms is established, by means of some properties involving the theory of Jacobi polynomials. The main result of the paper is based on some principles of Functional Analysis, and emphasizes the phenomenon of double condensation of singularities with respect to the considered interpolatory product quadrature formulas, by pointing out large subsets (in topological meaning) of the considered Banach spaces, on which the quadrature procedures are unboundedly divergent. PubDate: 2018-02-12 DOI: 10.1007/s10092-018-0253-8 Issue No:Vol. 55, No. 1 (2018)

Authors:Nguyen Buong; Pham Thi Thu Hoai Abstract: In this paper, we introduce implicit and explicit iterative methods for finding a zero of a monotone variational inclusion in Hilbert spaces. As consequence, an improvement modification of an algorithm existing in literature is obtained. A numerical example is given for illustrating our algorithm. PubDate: 2018-02-10 DOI: 10.1007/s10092-018-0250-y Issue No:Vol. 55, No. 1 (2018)

Authors:Davod Khojasteh Salkuyeh; Tahereh Salimi Siahkolaei Abstract: We introduce a two-parameter version of the two-step scale-splitting iteration method, called TTSCSP, for solving a broad class of complex symmetric system of linear equations. We present some conditions for the convergence of the method. An upper bound for the spectral radius of the method is presented and optimal parameters which minimize this bound are given. Inexact version of the TTSCSP iteration method (ITTSCSP) is also presented. Some numerical experiments are reported to verify the effectiveness of the TTSCSP iteration method and the numerical results are compared with those of the TSCSP, the SCSP and the PMHSS iteration methods. Numerical comparison of the ITTSCSP method with the inexact version of TSCSP, SCSP and PMHSS are presented. We also compare the numerical results of the BiCGSTAB method in conjunction with the TTSCSP and the ILU preconditioners. PubDate: 2018-02-10 DOI: 10.1007/s10092-018-0252-9 Issue No:Vol. 55, No. 1 (2018)

Authors:Petko D. Proinov Abstract: In this paper, we provide three types of general convergence theorems for Picard iteration in n-dimensional vector spaces over a valued field. These theorems can be used as tools to study the convergence of some particular Picard-type iterative methods. As an application, we present a new semilocal convergence theorem for the one-dimensional Newton method for approximating all the zeros of a polynomial simultaneously. This result improves in several directions the previous one given by Batra (BIT Numer Math 42:467–476, 2002). PubDate: 2018-02-09 DOI: 10.1007/s10092-018-0251-x Issue No:Vol. 55, No. 1 (2018)

Authors:Donatella Occorsio; Giada Serafini Abstract: The paper deals with the approximation of integrals of the type $$\begin{aligned} I(f;{\mathbf {t}})=\int _{{\mathrm {D}}} f({\mathbf {x}}) {\mathbf {K}}({\mathbf {x}},{\mathbf {t}}) {\mathbf {w}}({\mathbf {x}}) d{\mathbf {x}},\quad \quad {\mathbf {x}}=(x_1,x_2),\quad {\mathbf {t}}\in \mathrm {T}\subseteq \mathbb {R}^p, \ p\in \{1,2\} \end{aligned}$$ where \({\mathrm {D}}=[-\,1,1]^2\) , f is a function defined on \({\mathrm {D}}\) with possible algebraic singularities on \(\partial {\mathrm {D}}\) , \({\mathbf {w}}\) is the product of two Jacobi weight functions, and the kernel \({\mathbf {K}}\) can be of different kinds. We propose two cubature rules determining conditions under which the rules are stable and convergent. Along the paper we diffusely treat the numerical approximation for kernels which can be nearly singular and/or highly oscillating, by using a bivariate dilation technique. Some numerical examples which confirm the theoretical estimates are also proposed. PubDate: 2018-02-07 DOI: 10.1007/s10092-018-0243-x Issue No:Vol. 55, No. 1 (2018)

Authors:M. Akbas; L. G. Rebholz; C. Zerfas Abstract: We study a velocity–vorticity scheme for the 2D incompressible Navier–Stokes equations, which is based on a formulation that couples the rotation form of the momentum equation with the vorticity equation, and a temporal discretization that stably decouples the system at each time step and allows for simultaneous solving of the vorticity equation and velocity–pressure system (thus if special care is taken in its implementation, the method can have no extra cost compared to common velocity–pressure schemes). This scheme was recently shown to be unconditionally long-time \(H^1\) stable for both velocity and vorticity, which is a property not shared by any common velocity–pressure method. Herein, we analyze the scheme’s convergence, and prove that it yields unconditional optimal accuracy for both velocity and vorticity, thus making it advantageous over common velocity–pressure schemes if the vorticity variable is of interest. Numerical experiments are given that illustrate the theory and demonstrate the scheme’s usefulness on some benchmark problems. PubDate: 2018-02-07 DOI: 10.1007/s10092-018-0246-7 Issue No:Vol. 55, No. 1 (2018)

Authors:Long Chen; Jianguo Huang Abstract: Some error analyses on virtual element methods (VEMs) including inverse inequalities, norm equivalence, and interpolation error estimates are developed for polygonal meshes, each element of which admits a virtual quasi-uniform triangulation. This sub-mesh regularity covers the usual ones used for theoretical analysis of VEMs, and the proofs are presented by means of standard technical tools in finite element methods. PubDate: 2018-02-07 DOI: 10.1007/s10092-018-0249-4 Issue No:Vol. 55, No. 1 (2018)