Abstract: Publication date: Available online 14 October 2017 Source:Computers & Mathematics with Applications Author(s): K. Harish Kumar, V. Antony Vijesh In this paper, an iterative method based on quasilinearization is presented to solve a class of two dimensional partial integro differential equations that arise in nuclear reactor models and population models. Two different approaches based on Haar and Legendre wavelets are studied to develop numerical methods. In the first approach, time domain is approximated with the help of forward finite difference approach. In the second approach, both time as well as space domains are approximated by wavelets. Appropriate examples are solved using these methods and the obtained results are compared with the methods available in the recent literature.

Abstract: Publication date: Available online 14 October 2017 Source:Computers & Mathematics with Applications Author(s): Stefan Brechtken, Thomas Sasse This paper aims at approximations of the collision operator in the Boltzmann equation. The developed framework guarantees the “normality” of the approximation, which means correct collision invariants, H-Theorem, and equilibrium solutions. It fits into the discrete velocity model framework, is given in such a way that it is understandable with undergraduate level mathematics and can be used to construct approximations with arbitrary high convergence orders. At last we give an example alongside a numerical verification. Here the convergence orders range up to 3 ( 2 ) and the time complexity is given by 3 + 1 2 ( 4 + 2 3 ) in 2 ( 3 ) dimensions.

Abstract: Publication date: Available online 13 October 2017 Source:Computers & Mathematics with Applications Author(s): Fenglin Huang, Zhong Zheng, Yucheng Peng This work is devoted to investigate the spectral approximation of optimal control of parabolic problems. The space–time method is used to boost high-order accuracy by applying dual Petrov–Galerkin spectral scheme in time and spectral method in space. The optimality conditions are derived, and the a priori error estimates indicate the convergence of the proposed method. Numerical tests confirm the theoretical results, and show the efficiency of the method.

Abstract: Publication date: Available online 13 October 2017 Source:Computers & Mathematics with Applications Author(s): Liangliang Ma The magneto-micropolar fluid flows describe the motion of electrically conducting micropolar fluids in the presence of a magnetic field. The issue of whether the strong solution of magneto-micropolar equations in three-dimensional can exist globally in time with large initial data is still unknown. In this paper, we deal with the Cauchy problem of the three-dimensional magneto-micropolar system with mixed partial dissipation, magnetic diffusion and angular viscosity. More precisely, the global existence of smooth solutions to the three-dimensional incompressible magneto-micropolar fluid equations with mixed partial dissipation, magnetic diffusion and angular viscosity are obtained by energy method under the assumption that H 1 -norm of the initial data ( u 0 , b 0 , w 0 ) sufficiently small, namely ‖ u 0 , b 0 , ω 0 ‖ H 1 ( R 3 ) 2 ≤ ε 2 , where ε is a sufficiently small positive number. This work follows the techniques in the paper of Cao and Wu (2011).

Abstract: Publication date: Available online 10 October 2017 Source:Computers & Mathematics with Applications Author(s): Xiaowei Liu, Jin Zhang In this paper, we consider a singularly perturbed convection–diffusion equation posed on the unit square, where the solution has two characteristic layers and an exponential layer. A Galerkin finite element method on a Shishkin mesh is used to solve this problem. Its bilinear forms in different subdomains are carefully analyzed by means of a series of integral inequalities; a delicate analysis for the characteristic layers is needed. Based on these estimations, we prove supercloseness bounds of order 3 ∕ 2 (up to a logarithmic factor) on triangular meshes and of order 2 (up to a logarithmic factor) on hybrid meshes respectively. The result implies that the hybrid mesh, which replaces the triangles of the Shishkin mesh by rectangles in the exponential layer region, is superior to the Shishkin triangular mesh. Numerical experiments illustrate these theoretical results.

Abstract: Publication date: Available online 9 October 2017 Source:Computers & Mathematics with Applications Author(s): Jin Li, Hongxing Rui In this article, we discuss the classical composite trapezoidal rule for the computation of two dimensional singular integrals. The purpose is to obtain the convergence results O ( h 2 ) which is the same as the Riemann integral convergence rate at certain points of the classical composite trapezoidal rule. With the error functional of trapezoidal rule for computing two dimensional singular integrals, we get the superconvergence phenomenon when the special function in error functional is equal to zero. At last, some numerical examples are reported to illustrate our theoretical analysis which agree with it very well.

Abstract: Publication date: Available online 6 October 2017 Source:Computers & Mathematics with Applications Author(s): Hiba Fareed, John R. Singler, Yangwen Zhang, Jiguang Shen We propose an incremental algorithm to compute the proper orthogonal decomposition (POD) of simulation data for a partial differential equation. Specifically, we modify an incremental matrix SVD algorithm of Brand to accommodate data arising from Galerkin-type simulation methods for time dependent PDEs. The algorithm is applicable to data generated by many numerical methods for PDEs, including finite element and discontinuous Galerkin methods. The algorithm initializes and efficiently updates the dominant POD eigenvalues and modes during the time stepping in a PDE solver without storing the simulation data. We prove that the algorithm without truncation updates the POD exactly. We demonstrate the effectiveness of the algorithm using finite element computations for a 1D Burgers’ equation and a 2D Navier–Stokes problem.

Abstract: Publication date: Available online 6 October 2017 Source:Computers & Mathematics with Applications Author(s): Liejun Shen The present study is concerned with the nontrivial solutions for fractional Schrödinger–Poisson system with the Bessel operator. Under certain assumptions on the nonlinearity f , a nontrivial nonnegative solution is obtained by perturbation method for the given problem. In particular, the Ambrosetti–Rabinowitz type condition or the monotone assumption on the nonlinearity is unnecessary.

Abstract: Publication date: Available online 6 October 2017 Source:Computers & Mathematics with Applications Author(s): L. Shangerganesh, N. Nyamoradi, V.N. Deiva Mani, S. Karthikeyan The main goal of the present paper is establishing the existence and uniqueness of weak solutions for the nonlinear degenerate reaction–diffusion system with variable exponents. A model also is proposed to characterize the invasion of cancer cells towards healthy cells with acidification environment. Moreover, the main results of this paper are obtained using regularization problem, the Faedo–Galerkin approximation method, some apriori estimates, compactness results and the Gronwall Lemma.

Abstract: Publication date: Available online 6 October 2017 Source:Computers & Mathematics with Applications Author(s): Yijun He, Huaihong Gao, Hua Wang We consider the pseudo-parabolic p -Laplacian equation u t − Δ u t − Δ p u = u q − 2 u log ( u ) in a bounded domain with homogeneous Dirichlet boundary conditions. When 2 < p < q < p ( 1 + 2 n ) , we obtain results of the decay and the finite time blow-up for weak solutions.

Abstract: Publication date: Available online 6 October 2017 Source:Computers & Mathematics with Applications Author(s): Reza Ansari, Jalal Torabi, Ramtin Hassani The buckling analysis of thick composite annular sector plates reinforced with functionally graded carbon-nanotubes (CNTs) is presented under in-plane and shear loadings based on the higher-order shear deformation theory. It is considered that the plate is resting on the Pasternak-type elastic foundation. The overall material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are estimated through the micromechanical model. The governing equations are derived on the basis of the higher-order shear deformation plate theory, and the quadratic form of the energy functional of the system is presented. An efficient numerical method is presented in the context of variational formulation to obtain the discretized version of stability equations. The validation of the present study is demonstrated through comparisons with the results available in the literature and then comprehensive numerical results are given to investigate the impacts of model parameters on the stability of CNT-reinforced composite annular sector plates.

Abstract: Publication date: Available online 4 October 2017 Source:Computers & Mathematics with Applications Author(s): S.A. Hosseini, A. Eshghinejadfard, N. Darabiha, D. Thévenin Given the complex geometries usually found in practical applications, the Lattice Boltzmann (LB) method is becoming increasingly attractive for flow simulations. In addition to the simple treatment of intricate geometrical configurations, LB solvers can be implemented on very large parallel clusters with excellent scalability. However, reacting flows lead to additional challenges and have seldom been studied by LB methods. In this study, an in-house low Mach number Lattice Boltzmann solver, ALBORZ, has been extended to take into account multiple chemical components and reactions. For this purpose, the temperature and species of each mass fraction field are modeled through separate distribution functions. The flow distribution function is assumed to be independent of temperature and species mass fractions, which is valid in the limit of weak density variations. In order to compute reaction terms as well as variable thermodynamic and transport properties, the LB code has been coupled to another library of our group, REGATH. In this manner, LB simulations with detailed chemical kinetics and thermo-chemical models become possible. Since the code is currently limited to weak density variation, its performance has been checked for a laminar premixed as well as non-premixed counter-flow Ozone/Air reacting flow, describing kinetics with 4 species and 18 elementary reactions. Comparisons of the obtained reacting flow structures with results from classical finite-difference simulations show excellent agreement.

Abstract: Publication date: Available online 3 October 2017 Source:Computers & Mathematics with Applications Author(s): Ruming Zhang, Jiguang Sun, Chunxiong Zheng The reconstruction of a penetrable obstacle embedded in a periodic waveguide is a challenging problem. In this paper, the inverse problem is formulated as an optimization problem. We prove some properties of the scattering operator and propose an iterative scheme to approximate the support of the obstacle. Using the limiting absorption principle and a recursive doubling technique, we implement a fast algorithm based on a carefully designed finite element method for the forward scattering problem. Numerical examples validate the effectiveness of the method.

Abstract: Publication date: Available online 3 October 2017 Source:Computers & Mathematics with Applications Author(s): Leszek Demkowicz, Rick Falk, Benqi Guo, Michael Vogelius, Zhimin Zhang

Abstract: Publication date: 1 October 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 7 Author(s): Giuseppe Balduzzi, Simone Morganti, Ferdinando Auricchio, Alessandro Reali The present paper combines an effective beam theory with a simple and accurate numerical technique opening the door to the prediction of the structural behavior of planar beams characterized by a continuous variation of the cross-section geometry, that in general deeply influences the stress distribution and, therefore, leads to non-trivial constitutive relations. Accounting for these peculiar aspects, the beam theory is described by a mixed formulation of the problem represented by six linear Ordinary Differential Equations (ODEs) with non-constant coefficients depending on both the cross-section displacements and the internal forces. Due to the ODEs’ complexity, the solution can be typically computed only numerically also for relatively simple geometries, loads, and boundary conditions; however, the use of classical numerical tools for this problem, like a (six-field) mixed finite element approach, might entail several issues (e.g., shear locking, ill-conditioned matrices, etc.). Conversely, the recently proposed isogeometric collocation method, consisting of the direct discretization of the ODEs in strong form and using the higher-continuity properties typical of spline shape functions, allows an equal order approximation of all unknown fields, without affecting the stability of the solution. This makes such an approach simple, robust, efficient, and particularly suitable for solving the system of ODEs governing the non-prismatic beam problem. Several numerical experiments confirm that the proposed mixed isogeometric collocation method is actually cost-effective and able to attain high accuracy.

Abstract: Publication date: 1 October 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 7 Author(s): P. Constantinou, C. Xenophontos We consider a fourth order singularly perturbed boundary value problem posed in a square and the approximation of its solution by the h p version of the finite element method on the so-called Spectral Boundary Layer mesh. We show that the method converges uniformly, with respect to the singular perturbation parameter, at an exponential rate when the error is measured in the energy norm. Our theoretical findings are illustrated through a numerical example.

Abstract: Publication date: 1 October 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 7 Author(s): M. Faustmann, J.M. Melenk The h p -version of the finite element method is applied to singularly perturbed reaction–diffusion type equations on polygonal domains. The solution exhibits boundary layers as well as corner layers. On a class of meshes that are suitably refined near the boundary and the corners, robust exponential convergence (in the polynomial degree) is shown in both a balanced norm and the maximum norm.

Abstract: Publication date: 1 October 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 7 Author(s): Tom Gustafsson, Harri Hakula, Matti Leinonen We consider the approximation of the Reynolds equation with an uncertain film thickness. The resulting stochastic partial differential equation is solved numerically by the stochastic Galerkin finite element method with high-order discretizations both in spatial and stochastic domains. We compute the pressure field of a journal bearing in various numerical examples that demonstrate the effectiveness and versatility of the approach. The results suggest that the stochastic Galerkin method is capable of supporting design when manufacturing imperfections are the main sources of uncertainty.

Abstract: Publication date: 1 October 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 7 Author(s): Christoph Hofer In this paper we investigate the parallelization of dual–primal isogeometric tearing and interconnecting (IETI-DP) type methods for solving large-scale continuous and discontinuous Galerkin systems of equations arising from Isogeometric analysis of elliptic boundary value problems. These methods are extensions of the finite element tearing and interconnecting methods to isogeometric analysis. The algorithms are implemented by means of energy minimizing primal subspaces. We discuss how these methods can efficiently be parallelized in a distributed memory setting. Weak and strong scaling studies presented for two and three dimensional problems show an excellent parallel efficiency.

Abstract: Publication date: 1 October 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 7 Author(s): Bert Jüttler, Stefan K. Kleiss The present paper studies adaptive refinement on multi-patch domains in isogeometric analysis. In particular, we investigate the gluing construction for adaptively refined spline spaces to obtain discretizations that are C 0 smooth across interfaces. We will see that this is closely related to the concept of boundary compatibility of an adaptive spline construction. Given a spline basis (or, more generally, a generating system if linear independence is not guaranteed) on a d -dimensional box domain, there are two possibilities for constructing the spline basis on the domain boundary. Firstly, one can simply restrict the basis functions to the boundary. Secondly, one may restrict the underlying mesh to the boundary and construct the spline basis on the resulting mesh. The two constructions do not necessarily produce the same set of functions. If they do, then the spline bases are said to be compatible. We study this property for hierarchical (HB-) and truncated hierarchical B-splines (THB-splines) and identify sufficient conditions. These conditions are weaker for THB- than for HB-splines. Finally we demonstrate the importance of boundary compatibility for geometric modeling and for adaptive refinement in isogeometric analysis, in particular when considering multi-patch domains.

Abstract: Publication date: 1 October 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 7 Author(s): Yicong Lai, Yongjie Jessica Zhang, Lei Liu, Xiaodong Wei, Eugene Fang, Jim Lua Isogeometric analysis (IGA) has been developed for more than a decade. However, the usage of IGA is by far limited mostly within academic community. The lack of automatic or semi-automatic software platform of IGA is one of the main bottlenecks that prevent IGA from wide applications in industry. In this paper, we present a comprehensive IGA software platform that allows IGA to be incorporated into existing commercial software such as Abaqus, heading one step further to bridge the gap between design and analysis. The proposed IGA software framework takes advantage of user-defined elements in Abaqus, linking with general. IGES files from commercial computer aided design packages, Rhino specific files and mesh data. The platform includes all the necessary modules of the design-through-analysis pipeline: pre-processing, surface and volumetric T-spline construction, analysis and post-processing. Several practical application problems are studied to demonstrate the capability of the proposed software platform.

Abstract: Publication date: 1 October 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 7 Author(s): Jan Petsche, Andreas Schröder In this paper, mixed and mixed-hybrid methods for h - and h p -adaptive finite elements on quadrilateral meshes are discussed for variational equations and, in particular, for variational inequalities. The main result is the derivation of reliable error estimates for mixed methods for the obstacle problem. The estimates rely on the use of a post-processing of the potential in H 1 and on the introduction of a certain Lagrange multiplier which is associated with the obstacle constraints. The error estimates consist of the dual norm of the residual, which is defined by an appropriate approximation of the Lagrange multiplier, plus some computable remainder terms. In numerical experiments, the applicability of the post-processing procedure on quadrilateral meshes with multilevel hanging-nodes is verified and the use of the estimates in h - and h p -adaptive schemes is demonstrated by means of convergence rates and effectivity indices.

Abstract: Publication date: 1 October 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 7 Author(s): Lars Radtke, Marcel König, Alexander Düster We present a study of the fluid–structure interaction in an idealized end-to-end anastomosis of a vascular bypass-graft and an artery. Special attention is paid to the impact of geometric imperfections in the artery and the flow path of the upstream vessel segment on the hemodynamics. A partitioned solution approach is applied and developed further to solve the coupled problem in an implicit manner. To stabilize and accelerate the convergence of the staggered coupling iterations, an interface quasi-Newton least squares method is applied. While the finite volume method is used for the fluid mechanics subproblem, high-order finite elements serve to discretize the structural subproblem. A convergence study shows the efficiency of the high-order elements in the context of nearly incompressible, anisotropic materials used to model circular and irregular-shaped segments of an artery. The fluid–structure interaction simulations reveal a dominant influence of the upstream vessel’s curvature, which, however, decays rapidly in straight sections where the influence of geometric imperfections is dominant. Based on the proposed simulation approach, hemodynamic parameters such as the oscillating shear index can be directly linked to the shape and the intensity of the imperfections.

Abstract: Publication date: 1 October 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 7 Author(s): Benjamin Wassermann, Stefan Kollmannsberger, Tino Bog, Ernst Rank During the last ten years, increasing efforts were made to improve and simplify the process from Computer Aided Design (CAD) modeling to a numerical simulation. It has been shown that the transition from one model to another, i.e. the meshing, is a bottle-neck. Several approaches have been developed to overcome this time-consuming step, e.g. Isogeometric Analysis (IGA), which applies the shape functions used for the geometry description (typically B-Splines and NURBS) directly to the numerical analysis. In contrast to IGA, which deals with boundary represented models (B-Rep), our approach focuses on parametric volumetric models such as Constructive Solid Geometries (CSG). These models have several advantages, as their geometry description is inherently watertight and they provide a description of the models’ interior. To be able to use the explicit mathematical description of these models, we employ the Finite Cell Method (FCM). Herein, the only necessary input is a reliable statement whether an (integration-) point lies inside or outside of the geometric model. This paper mainly discusses such point-in-membership tests on various geometric objects like sweeps and lofts, as well as several geometric operations such as filleting or chamfering. We demonstrate that, based on the information of the construction method of these objects, the point-in-membership-test can be carried out efficiently and robustly.

Abstract: Publication date: 1 October 2017 Source:Computers & Mathematics with Applications, Volume 74, Issue 7 Author(s): Adam Zdunek, Waldemar Rachowicz A higher order mixed finite element method is presented for compressible transversely isotropic finite hyperelasticity. The independent variables of the three-field formulation are; displacement, fibre tension and fibre stretch. The formulation admits the description of a fully coupled stress response which evolves from an almost isotropic one into a hyper-anisotropic simply nearly inextensible response with increasing fibre tension, as for example observed in soft tissue biomechanics. Standard displacement approximation of order p in H 1 ( Ω ) is used while the auxiliary variables are approximated element wise by square integrable functions of order p − 1 in L 2 ( Ω ) . For finite extensibility the auxiliary variables are statically condensed out yielding a pure displacement based method. It is implemented in an hp-adaptive finite element code. A matching residual based error estimation capability is added and exercised. Numerical evidence indicating stability of approximation is supplied resolving a boundary layer caused by almost inextensible fibres. Coupled and uncoupled stress responses are compared. A generalised compressible transversely isotropic Holzapfel–Gasser–Ogden model is developed for the formulation that intentionally avoids the volumetric–isochoric split. Solutions obtained with the mixed method compare favourably to the corresponding ones obtained with pure displacement formulation. The latter fails in certain strongly anisotropic cases.

Abstract: Publication date: Available online 29 September 2017 Source:Computers & Mathematics with Applications Author(s): Maxence Cassier, Patrick Joly, Maryna Kachanovska In this work, we investigate mathematical models for electromagnetic wave propagation in dispersive isotropic media. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notions of non-dissipativity and passivity. We consider successively the cases of so-called local media and then of general passive media. The models are studied through energy techniques, spectral theory and dispersion analysis of plane waves. For making the article self-contained, we provide in appendix some useful mathematical background.

Abstract: Publication date: Available online 29 September 2017 Source:Computers & Mathematics with Applications Author(s): Congyin Fan, Kaili Xiang, Shanzhen Chen The empirical test suggests that the log-return series of stock price in US market reject the normal distribution and admit instead a subclass of the asymmetric distribution. In this paper, we investigate the stock loan problem under the assumption that the return of stock follows the finite moment log-stable process (FMLS). In this case, the pricing model of stock loan can be described by a space-fractional partial differential equation with time-varying free boundary condition. Firstly, a penalty term is introduced to change the original problem to be defined on a fixed domain, and then a fully-implicit difference scheme has been developed. Secondly, based on the fully-implicit scheme, we prove that the stock loan value generated by the penalty method cannot fall below the value obtained when the stock loan is exercised early. Thirdly, the numerical experiments are carried out to demonstrate differences of stock loan model under the FMLS and the standard normal distribution. Optimal redemption strategy of stock loan has been achieved. Furthermore the impact of key parameters in our model on the stock loan evaluation are analyzed, and some reasonable explanation are given.

Abstract: Publication date: Available online 29 September 2017 Source:Computers & Mathematics with Applications Author(s): Jun Liu, Zhu Wang Optimization with time-dependent partial differential equations (PDEs) as constraints appears in many science and engineering applications. The associated first-order necessary optimality system consists of one forward and one backward time-dependent PDE coupled with optimality conditions. An optimization process by using the one-shot method determines the optimal control, state and adjoint state at once, with the cost of solving a large scale, fully discrete optimality system. Hence, such a one-shot method could easily become computationally prohibitive when the time span is long or time step is small. To overcome this difficulty, we propose several time domain decomposition algorithms for improving the computational efficiency of the one-shot method. In these algorithms, the optimality system is split into many small subsystems over a much smaller time interval, which are coupled by appropriate continuity matching conditions. Both one-level and two-level multiplicative and additive Schwarz algorithms are developed for iteratively solving the decomposed subsystems in parallel. In particular, the convergence of the one-level, non-overlapping algorithms is proved. The effectiveness of our proposed algorithms is demonstrated by both 1D and 2D numerical experiments, where the developed two-level algorithms show convergence rates that are scalable with respect to the number of subdomains.

Abstract: Publication date: Available online 29 September 2017 Source:Computers & Mathematics with Applications Author(s): Jiayang Wu, Yongguang Cheng, Chunze Zhang, Wei Diao All explicit immersed boundary–lattice Boltzmann (IB–LB) coupling schemes suffer from instability and restrictive choices of timestep and boundary rigidity. In this paper, an implicit IB–LB coupling method that can greatly alleviate those problems and easily incorporate the boundary mass is presented. The nonlinear equations resulting from the implicit discretization of the immersed boundary (IB) equations are solved by the Jacobian-free Newton–Krylov (JFNK) method, while the boundary mass is handled by introducing ghost boundaries. The improvement in stability is verified by two benchmark cases, and it is evident that both the maximal timestep and bending rigidity attained by the proposed implicit method can increase up to 200 times compared with those of the explicit version. The new method is applied to analyzing the vortex induced vibration of an impulsively started flexible filament, and both the simulated flow field and filament motion agree well with those obtained by particle image velocimetry (PIV). The influences of filament parameters on the vibration characteristics are discussed, and according to the results, the filament mass is the main influencing factor of vibration amplitude and the bending rigidity is the main influencing factor of vibration frequency.

Abstract: Publication date: Available online 29 September 2017 Source:Computers & Mathematics with Applications Author(s): Andrea M.P. Valli, Regina C. Almeida, Isaac P. Santos, Lucia Catabriga, Sandra M.C. Malta, Alvaro L.G.A. Coutinho In this paper, we present a two-scale finite element formulation, named Dynamic Diffusion (DD), for advection–diffusion–reaction problems. By decomposing the velocity field in coarse and subgrid scales, the latter is used to determine the smallest amount of artificial diffusion to minimize the coarse-scale kinetic energy. This is done locally and dynamically, by imposing some constraints on the resolved scale solution, yielding a parameter-free consistent method. The subgrid scale space is defined by using bubble functions, whose degrees of freedom are locally eliminated in favor of the degrees of freedom that live on the resolved scales. Convergence tests on a two-dimensional example are reported, yielding optimal rates. In addition, numerical experiments show that DD method is robust for a wide scope of application problems.

Abstract: Publication date: Available online 29 September 2017 Source:Computers & Mathematics with Applications Author(s): Kexue Li This paper is concerned with a class of semilinear stochastic delayed reaction–diffusion equations driven by Lévy noise in a separable Hilbert space. We establish sufficient conditions to ensure the existence of a unique positive solution. Moreover, we study blow-up of solutions in finite time in mean L p -norm sense. Several examples are given to illustrate applications of the theory.

Abstract: Publication date: Available online 28 September 2017 Source:Computers & Mathematics with Applications Author(s): Jinhong Jia, Hong Wang Fractional partial differential equations (PDEs) provide a powerful and flexible tool for modeling challenging phenomena including anomalous diffusion processes and long-range spatial interactions, which cannot be modeled accurately by classical second-order diffusion equations. However, numerical methods for space-fractional PDEs usually generate dense or full stiffness matrices, for which a direct solver requires O ( N 3 ) computations per time step and O ( N 2 ) memory, where N is the number of unknowns. The significant computational work and memory requirement of the numerical methods makes a realistic numerical modeling of three-dimensional space-fractional diffusion equations computationally intractable. Fast numerical methods were previously developed for space-fractional PDEs on multidimensional rectangular domains, without resorting to lossy compression, but rather, via the exploration of the tensor-product form of the Toeplitz-like decompositions of the stiffness matrices. In this paper we develop a fast finite difference method for distributed-order space-fractional PDEs on a general convex domain in multiple space dimensions. The fast method has an optimal order storage requirement and almost linear computational complexity, without any lossy compression. Numerical experiments show the utility of the method.

Abstract: Publication date: Available online 28 September 2017 Source:Computers & Mathematics with Applications Author(s): Pablo Pedregal We introduce a heuristic, practical procedure to take into account, in an easy-to-implement way, point-wise constraints in a variational problem in several dimensions. In addition to showing a convergence result under suitable assumptions, we emphasize the flexibility of the method by using it for an optimal design problem either in conductivity or elasticity, where an optimal mixture of two materials is to be found in a given design domain, under pointwise constraints. We illustrate the performance of the algorithm with some simple test examples.

Abstract: Publication date: Available online 28 September 2017 Source:Computers & Mathematics with Applications Author(s): Sergey Repin, Stanislav Sysala, Jaroslav Haslinger We propose a new method for analyzing the limit (safe) load of elastoplastic media governed by the Hencky plasticity law and deduce fully computable bounds of this load. The main idea of the method is based on a combination of kinematic approach and new estimates of the distance to the set of divergence free fields. We show that two sided bounds of the limit load are sharp and the computational efficiency of the method is confirmed by numerical experiments.

Abstract: Publication date: Available online 28 September 2017 Source:Computers & Mathematics with Applications Author(s): Manju Bisht, Dhiraj V. Patil In this article, the flow of non-Newtonian fluid (which is represented by the power-law model) in two-dimensional (2D) driven enclosures is studied. The enclosure consists of regular, rectangular shaped undulations on the bottom wall. Multiple-Relaxation time (MRT) collision model for the lattice Boltzmann equation method (LBM) is employed. First, numerical validation is performed by comparing the MRT-LBM results of power-law fluid flow inside the wall-driven square enclosure (no undulation) and flow inside diagonally flipped L-shaped enclosure with the literature. The strain rate profiles for square enclosure without undulations are compared using various equations available for strain rate calculation in the LBM literature. Further, the effect of different values of the non-hydrodynamic relaxation parameters on the flow is examined. Then, for the undulated enclosures, flow features and eddy dynamics are analyzed and discussed for the variations in the power-law index, n , to represent shear-thinning and shear thickening fluids. The effects of various parameters such as Reynolds numbers, wall undulation heights and wavelength of undulations on the power-law fluid flow are analyzed. Also, the variation of viscosity with spatial location for steady-state flow and total kinetic energy within the computational domain are presented for various values of power-law index.

Abstract: Publication date: Available online 27 September 2017 Source:Computers & Mathematics with Applications Author(s): Biao Zeng, Stanisław Migórski In this paper we consider the first order evolutionary inclusions with nonlinear weakly continuous operators and a multivalued term which involves the Clarke subgradient of a locally Lipschitz function. First, we provide a surjectivity result for stationary inclusion with weakly–weakly upper semicontinuous multifunction. Then, we use this result to prove the existence of solutions to the Rothe sequence and the evolutionary subgradient inclusion. Finally, we apply our results to the non-stationary Navier–Stokes equation with nonmonotone and multivalued frictional boundary conditions.

Abstract: Publication date: Available online 25 September 2017 Source:Computers & Mathematics with Applications Author(s): Zhixing Fu, Norbert Heuer, Francisco-Javier Sayas We propose and analyze a new coupling procedure for the Hybridizable Discontinuous Galerkin Method with Galerkin Boundary Element Methods based on a double layer potential representation of the exterior component of the solution of a transmission problem. We show a discrete uniform coercivity estimate for the non-symmetric bilinear form and prove optimal convergence estimates for all the variables, as well as superconvergence for some of the discrete fields. Some numerical experiments support the theoretical findings.

Abstract: Publication date: Available online 25 September 2017 Source:Computers & Mathematics with Applications Author(s): Armin Lechleiter, Ruming Zhang Scattering of non-periodic waves from unbounded structures is difficult to treat, as one typically formulates the problem in an unbounded domain covering the unbounded periodic structure. The Floquet–Bloch transform reduces the latter problem to a family of decoupled periodic scattering problems. This reduction is in particular interesting from the point of view of numerical computations. We analyze a corresponding fully discrete numerical solution algorithm for three-dimensional scattering problems in acoustics and electromagnetics, proving in particular convergence rates under suitable assumptions on the geometry and the material coefficients. A crucial part of our analysis relies on the continuous dependence of the family of solutions to the quasiperiodic scattering problems on the quasiperiodicity. The latter part is actually more difficult to establish than for corresponding two-dimensional problems. We further provide a numerical example in 3D acoustics that illustrates feasibility of the proposed algorithm.

Abstract: Publication date: Available online 25 September 2017 Source:Computers & Mathematics with Applications Author(s): Amiya Das, Niladri Ghosh, Khusboo Ansari In this article, we introduce the dual power Zakharov–Kuznetsov–Burgers equation with fractional temporal evolution in the sense of modified Riemann–Liouville derivative. We investigate the dynamical behavior, bifurcations and phase portrait analysis of the exact traveling wave solutions of the system with and without damping effect. We apply the ( G ′ ∕ G ) -expansion method in context of fractional complex transformation and seek a variety of exact traveling wave solutions including solitary wave, kink-type wave, breaking wave and periodic wave solutions of the equation. Furthermore, the remarkable features of the traveling wave solutions and phase portraits of dynamical system are demonstrated through interesting figures.

Abstract: Publication date: Available online 25 September 2017 Source:Computers & Mathematics with Applications Author(s): Yingwen Guo, Yinnian He In this paper, we study stability and convergence of fully discrete finite element method on large timestep which used Crank–Nicolson extrapolation scheme for the nonstationary Navier–Stokes equations. This approach bases on a finite element approximation for the space discretization and the Crank–Nicolson extrapolation scheme for the time discretization. It reduces nonlinear equations to linear equations, thus can greatly increase the computational efficiency. We prove that this method is unconditionally stable and unconditionally convergent. Moreover, taking the negative norm technique, we derive the L 2 , H 1 -unconditionally optimal error estimates for the velocity, and the L 2 -unconditionally optimal error estimate for the pressure. Also, numerical simulations on unconditional L 2 -stability and convergent rates of this method are shown.

Abstract: Publication date: Available online 22 September 2017 Source:Computers & Mathematics with Applications Author(s): Mukesh Kumar, Dig Vijay Tanwar, Raj Kumar In the present research, similarity transformation method via Lie-group theory is proposed to seek some more exact closed form solutions of the (2+1)-dimensional breaking soliton system. The system describes the interactions of the Riemann wave along y-axis and long wave along x-axis. Some explicit solutions of breaking soliton system are attained with appropriate choices of the arbitrary functions and making use of arbitrary constants involved in the infinitesimals. In order to obtain physically meaningful solutions, numerical simulation is performed. On the basis of graphical representation, the physical analysis of solutions reveals into multi-solitons, periodic, quadratic, asymptotic and stationary profiles.

Abstract: Publication date: Available online 22 September 2017 Source:Computers & Mathematics with Applications Author(s): Constantin Bacuta, Fioralba Cakoni, Houssem Haddar, Jiguang Sun

Abstract: Publication date: Available online 20 September 2017 Source:Computers & Mathematics with Applications Author(s): Jianhua Zhang, Jing Zhao For generalized saddle point problems, we establish a new matrix splitting preconditioner and give the implementing process in detail. The new preconditioner is much easier to be implemented than the modified dimensional split (MDS) preconditioner. The convergence properties of the new splitting iteration method are analyzed. The eigenvalue distribution of the new preconditioned matrix is discussed and an upper bound for the degree of its minimal polynomial is derived. Finally, some numerical examples are carried out to verify the effectiveness and robustness of our preconditioner on generalized saddle point problems discretizing the incompressible Navier–Stokes equations.

Abstract: Publication date: Available online 20 September 2017 Source:Computers & Mathematics with Applications Author(s): Hongwei Zhang, Qingying Hu In this paper we consider the initial boundary value problem for a class of fractional logarithmic Schrödinger equation. By using the fractional logarithmic Sobolev inequality and introducing a family of potential wells, we give some properties of the family of potential wells and obtain existence of global solution.