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The author considers the application of the approximation theory methods to the principles of optimality in the decision-making theory. In finding optimal solutions, the risk function often has rather complex structure for studying its properties, which makes it necessary to approximate the risk function to another function with simple and clear characteristics. In this regard, the asymptotic properties of the solutions of biharmonic equations as approximate functions are investigated. Complete asymptotic expansions of the upper limits of deviations of the Sobolev class functions W2 (the set that the risk functions in decision-making optimization belong to) from operators that are solutions of biharmonic equations with certain boundary conditions are obtained. The expansions allow us to find the Kolmogorov–Nikolsky constants of arbitrarily high degree of smallness, which makes it possible to estimate the approximation error when solving optimization problems with arbitrary accuracy. It is mentioned that the biharmonic equations can be used to efficiently generate mathematical models of natural and social phenomena. [ABSTRACT FROM AUTHOR]

We study analogues of the Rayleigh waves in a semiinfinite Kirchhoff plate with periodic edges (the Neumann problem for the biharmonic operator) and in an infinite plate with a periodic family of foreign inclusions. In the first case, we prove the existence of localized waves (exponentially decaying in the perpendicular direction) for any edge profile. In the second case, we obtain a sufficient condition for trapping of waves. We explain why the results obtained for the biharmonic operator are different from the known results for the Helmholtz equation. In the case of threshold resonance, we construct asymptotic expansions of eigenvalues of the model problem, which generate analogues of the Stoneley waves. For a plate with a periodic family of cracks perpendicular to the straight boundary of the half-plane, we detect periodic localized Floquet waves that do not transfer energy. [ABSTRACT FROM AUTHOR]

We study the uniqueness and the asymptotic expansion of the solution of the mixed Dirichlet–Neumann problem for the biharmonic equation in the exterior of a compact set under the assumption that the generalized solution of this problem has a finite Dirichlet integral with power-law weight with exponent . Depending on the value of the parameter , uniqueness (nonuniqueness) theorems are proved and exact formulas are found for the dimension of the linear solution space of the mixed Dirichlet–Neumann problem. [ABSTRACT FROM AUTHOR]

An attempt has been made to examine the two dimensional free stream squeezed flow through a horizontal sensor surface of an electrically conducting Carreau-Yasuda fluid subject to transverse magnetic field. The governing nonlinear partial differential equations are modeled and then simplified with the aid of suitable similarity transformations. Well known numerical scheme Runge–Kutta–Fehlberg method is utilized to solve the system. Concrete graphical analysis is carried out to investigate the behavior of different pertinent parameters on velocity profile with inclusive discussion. Numerical influence of friction factor is also discussed. [ABSTRACT FROM AUTHOR]

In the paper, we consider the Dirichlet boundary value problem for the biharmonic equation defined in a thin T-like shaped structure. Our goal is to construct an asymptotic expansion of its solution. We provide error estimates and also introduce and justify the asymptotic partial domain decomposition for this problem. [ABSTRACT FROM AUTHOR]

We consider the entire solution of the semilinear biharmonic equation Δ 2 u = e u , in R N , N ≥ 1 . For N = 3 , we obtain the asymptotic of the entire nonradial solution which extends the results of Lai and Ye (in press). Besides, a new singular solution of the above equation is constructed by derivative estimates for N ≥ 13 . [ABSTRACT FROM AUTHOR]

Abstract: Here we give a self-contained new proof of the asymptotic behavior of the radially symmetric entire solution of These results were obtained by I. Guerra in [4]. Our proof is much more direct and simpler. [Copyright &y& Elsevier]

Abstract: Using the estimates of the principal eigenvalue of the -biharmonic operator on the ball from below and from above, we provide its asymptotic analysis as . [Copyright &y& Elsevier]

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Abstract: In this paper we study radially symmetric entire solutions of We find the asymptotic behavior of these solutions for any and prove that for any there exists a solution with exactly linear growth. [Copyright &y& Elsevier]

In this paper, we research the standard 13-point difference scheme for solving the biharmonic equation. Existence and convergence of finite difference methods(FDM) solutions are obtained by estimating the lower bounds of the minimum eigenvalues of the discrete matrix and making use of the Taylor series expansions, respectively. The accuracy are proved to be O(h²). Moreover, the extrapolation techniques are used to improve the high accuracy of the solutions. For the given examples, numerical results show that the errors O(h4) is obtained from the first level extrapolation. The very accurate solutions with the errors O(10-8) are obtained by the third level extrapolation techniques under the homogeneous essential boundary conditions. [ABSTRACT FROM AUTHOR]

This paper presents a review of the numerical analysis of composite steel-concrete beams, with particular attention on the modelling of composite beams containing deep trapezoidal slabs cast onto profiled steel sheeting. It is concluded that while robust algorithms are available, there is considerable scope for improvement of these models as an alternative to undertaking expensive testing programs in order to formulate practical design procedures. This is of particular relevance to Australian practice, as timely revisions of its composite structures standard are needed [ABSTRACT FROM AUTHOR]

Cancellous bone can be regarded as a lattice of asymptotically small rods and plates. In this article, a thin plate of an isotropic Kelvin–Voigt viscoelastic material with a thickness ϵ ≪ 1 is considered. This thickness leads to asymptotic expansions of the displacement, stress and strain tensors, and of the change in the volumetric fraction of solid bone in the bone matrix. A rate equation describing the change in the volumetric fraction with respect to time governs the remodeling process of bone deposition and reabsorption. It has linear and quadratic terms of strain tensors and thus, its own asymptotic expansion. This expansion's leading term is used in some simple numerical simulations. [ABSTRACT FROM AUTHOR]

Abstract: By a variant version of Fountain Theorem due to Zou [W. Zou, Variant fountain theorems and their applications, Manuscripta Math. 104 (2001) 343–358], the existence of infinitely many solutions is obtained for a class of biharmonic equations where the nonlinearity involves a combination of superlinear and asymptotically linear terms. [Copyright &y& Elsevier]

In an arbitrary bounded 2-D domain, a singular perturbation approach is developed to analyze the asymptotic behavior of several biharmonic linear and nonlinear eigenvalue problems for which the solution exhibits a concentration behavior either due to a hole in the domain, or as a result of a nonlinearity that is nonnegligible only in some localized region in the domain. The specific form for the biharmonic nonlinear eigenvalue problem is motivated by the study of the steady-state deflection of one of the two surfaces in a Micro-Electro-Mechanical System capacitor. The linear eigenvalue problem that is considered is to calculate the spectrum of the biharmonic operator in a domain with an interior hole of asymptotically small radius. One key novel feature in the analysis of our singularly perturbed biharmonic problems, which is absent in related second-order elliptic problems, is that a point constraint must be imposed on the leading order outer solution to asymptotically match inner and outer representations of the solution. Our asymptotic analysis also relies heavily on the use of logarithmic switchback terms, notorious in the study of Low Reynolds number fluid flow, and on detailed properties of the biharmonic Green's function and its associated regular part near the singularity. For a few simple domains, full numerical solutions to the biharmonic problems are computed to verify the asymptotic results obtained from the analysis. [ABSTRACT FROM AUTHOR]

We introduce a residual-based a posteriori error indicator for discontinuous Galerkin discretizations of the biharmonic equation with essential boundary conditions. We show that the indicator is both reliable and efficient with respect to the approximation error measured in terms of a natural energy norm under minimal regularity assumptions. We validate the performance of the indicator within an adaptive mesh refinement procedure and show its asymptotic exactness for a range of test problems. [ABSTRACT FROM PUBLISHER]

Abstract: The Powell–Sabin (PS) piecewise quadratic finite element on the PS 12-split of a triangulation is a common choice for the construction of a BPX-type preconditioner for the biharmonic equation. In this note we investigate the related Powell–Sabin element on the PS 6-split instead of the PS 12-split for the construction of such preconditioners. For the PS 6-split element multilevel spaces can be created using a -refinement scheme instead of the traditional dyadic scheme. Topologically -refinement has many advantages: it is a slower refinement than the dyadic split operation, and it alternates the orientation of the refined triangles. Therefore we expect a reduction of the amount of work when compared to the PS 12-split element BPX preconditioner, although both methods have the same asymptotical complexity. Numerical experiments confirm this statement. [Copyright &y& Elsevier]

Abstract: The paper describes the development of an algorithm that uses spline interpolation methods in order to calculate the ground relief contours, the area and the volume reservoir associated with a large dam. This algorithm is a great help to the engineer who designs dams in calculating the area and the volume of the reservoir for dams with specific positions and geometric characteristics, for different water levels and over a ground relief with elevation contours which one evaluates, smoothes and draws using cubic splines. The advantages of the algorithm are significant not only for specific computer programs that refer to reservoirs, but also for computer programs relevant to hydraulic calculations of reservoirs and associated structures. [Copyright &y& Elsevier]

We study the asymptotic behavior of least energy solutions to the equation Δ2u=c0K(x)upℇ with the Navier boundary condition as ℇ→+0, where Ω is a smooth bounded domain in RN (N≥5), c0=(N-4)(N-2)N(N+2) and pℇ=(N+4)/(N-4)-ℇ, ℇ>0. Under some assumptions on the coefficient function K, we obtain fairly precise asymptotics of the L∞-norm of least energy solutions. [ABSTRACT FROM AUTHOR]

Abstract: The Cayley Transform, , with and , where denotes spectrum, is of significant theoretical importance and interest and has many practical applications. E.g., in the solution of the Linear Complementarity Problem (LCP), in the solution of linear systems arising from the discretization of model problems elliptic PDEs by Alternating Direction Implicit (ADI) iterative methods, in the solution of complex linear systems by ADI-type methods of Hermitian/Skew Hermitian or Normal/Skew Hermitian Splittings, etc. In the present work we apply the principle of Extrapolation to generalize the Cayley Transform and determine in an optimal sense the extrapolation parameter involved so that problems in many practical applications are solved much more efficiently. [Copyright &y& Elsevier]

Abstract: The asymptotic solution of the singular stress field near a singular point is generally comprised of one or more singular terms in the form of . Based on the asymptotic solution of the singular stress field and the common numerical solution (stresses or displacements) obtained by an ordinary tool such as the finite element method or boundary element method, a simple and effective numerical method is developed to calculate stress intensity factors for one and two singularities. Three examples show that the stress intensity factors evaluated using the method proposed in this paper are very accurate. [Copyright &y& Elsevier]

Abstract: We design diffractive micro-optical elements (DMOEs) with sub-wavelength characteristic size based on a half-π-phase shifting scheme, with the aim of realizing a hollow beam and bi-focus simultaneously. The optical properties of DMOEs, such as the positions of the real focal planes, the aspect ratio of the hollow beam and the diffraction efficiency, are investigated by using the rigorous electromagnetic theory and the boundary element method. The numerical results show that a well-defined hollow beam and bi-focus are achieved simultaneously. Further analysis of the interference between dual-DMOE shows that the interference is much less than that of a conventional dual-microlens. It is indicated that the designed micro-optical element array can not only implement hollow beam and dual focus, but also suppress the interference effect significantly. [Copyright &y& Elsevier]

Abstract: Using slope enclosures may provide sharper bounds for the range of a function than using enclosures of the derivative. Hence, slope enclosures may be useful in verifying the assumptions for existence tests or in algorithms for global optimization. Previous papers by Kolev and Ratz show how to compute slope enclosures for convex and concave functions. In this paper, we generalize these formulas and show how to obtain slope enclosures for a function that has exactly one point of inflection or whose derivative has exactly one point of inflection. [Copyright &y& Elsevier]

Abstract: In this paper, we point out that the generalized iteration method presented in a recent paper by Ling and Hu [X.-F. Ling, X.-Z. Hu, On the iterative algorithm for large sparse saddle point problems, Appl. Math. Comput. 178 (2006) 372–379] belongs to the inexact Uzawa algorithm, and an improper statement in a theorem in that paper is corrected. [Copyright &y& Elsevier]

Abstract: A numerical study of three-dimensional natural convection in an attic space with heated horizontal base and cooled upper walls is presented. Every previous study pertinent to this subject as of today has assumed that the flow in attics is two-dimensional and restricted to triangular cavities. This problem is examined for fixed aspect ratios holding width to height of 2 and depth to height of 3.33 and Rayleigh numbers ranging from 104 to 8×105. The coupled system of conservation equations, subject to the proper boundary conditions, along with the equation of state assuming the air behaves as a perfect gas are solved with the finite volume method. In the conservation equations, the second-order-accurate QUICK scheme was used for the discretization of the convective terms and the SIMPLE scheme for the pressure-velocity coupling. It is categorically found that the flow in the attic is 3D. From the physics of the problem, two steady-state solutions are possible. The symmetrical solution prevails for relatively low Rayleigh numbers. However, as the Ra is gradually increased, a transition occurs at a critical value Ra C. Above this value of Ra C, an asymmetrical solution exhibiting a pitchfork bifurcation arises and eventually becomes steady. Results are presented detailing the occurrence of the pitchfork bifurcation and the resulting flow patterns are described. [Copyright &y& Elsevier]

Abstract: The interaction between a submerged elastic circular cylindrical shell and an external shock wave is addressed. A linear, two-dimensional formulation of the problem is considered. A semi-analytical solution is obtained using a combination of the classical analytical approach based on the use of the Laplace transform and separation of variables, and finite difference methodology. The study consists of two parts. Part I focuses on the simulation and analysis of the acoustic fields induced during the interaction. Both the diffraction (absolutely rigid cylinder) and complete diffraction–radiation (elastic shell) are considered. Special attention is paid to the lower-magnitude shell-induced waves representing radiation by the elastic waves circumnavigating the shell. The focus of Part II is on the numerical analysis of the solution. The convergence of the series solution and finite-difference scheme is analysed. The computation of the response functions of the problem is discussed as well, as is the effect of the bending stiffness on the acoustic field. The membrane model of the shell is considered to analyse such an effect, which, in combination with the models addressed in Part I, allows for the analysis of the evolution of the acoustic field around the structure as its elastic properties change from an absolutely rigid cylinder to a membrane. The results of the numerical simulations are compared to available experimental data, and a good agreement is observed. [Copyright &y& Elsevier]

Abstract: Three-dimensional electrostatic weighting field configuration in a typical Resistive Plate Chamber (RPC) has been studied with the neBEM solver which evaluates accurate potential and field distributions in a given chamber following Boundary Element Method (BEM). The variation in three-dimensional weighting field of the RPC has been studied in detail with the change in design parameters like dimension, material, location of several components. The advantage of using the numerical solver neBEM in comparison to analytical solutions and other numerical approaches has been discussed in connection to the evaluation of weighting field. [Copyright &y& Elsevier]

Abstract  A unified method of approximation, extrapolation, and objective layering is offered for processing vertical oceanographic profiles. The method is demonstrated using seawater density and consists of adjustable splitting of each individual profile into N vertical layers based on tentative, piecewise linear homogeneous approximation with specified accuracy and a final fitting of an N-layered analytical model to data. A set of 3N coefficients of the model includes one density value at the sea surface; N−1 depths of layer interfaces; and N pairs of coefficients that describe a profile shape within the n-th layer—an asymptotic density value (a key parameter for extrapolation) and a vertical scale of maximum density variability (related to vertical gradient). Several distinctive characteristics of the technique are: (1) It can be used for the analysis of the vertical structure of individual profiles when N is an unknown parameter, and spatial interpolation when N should be equal for all profiles. (2) A justified downward extrapolation of incomplete data is possible with the model, especially if historical deepwater profiles are available. (3) Layer interfaces, as well as other coefficients, are derived with only one fitting to the entire profile. (4) The technique, using its general formulation, can serve as a parent for developing various types of models. The simpler step-like (with hyperbolic or exponential approximation) and more complicated smooth (continuous in gradient space) models were designed and tested against a large number of density profiles from the Sea of Okhotsk and the Gulf of California. Comparison of parametric, z-levels and isopycnal averaging was done for the region off the northeastern coast of Sakhalin. [ABSTRACT FROM AUTHOR]

Abstract: A class of discrete-time orthogonal spline collocation schemes for solving coupled Klein–Gordon–Schrödinger equations with initial and boundary conditions are considered. These schemes are constructed by using piecewise cubic Hermite interpolations in space combined with finite difference methods in time. It is proved that the schemes have the conservation laws of discrete energy, and possess second order accuracy in maximum norm and fourth order in -norm for time and space, respectively. The conservation laws and rate of convergence are verified in numerical experiments. Moreover, the propagations of solitary waves and collisions of two head-on solitary waves are also well simulated. [Copyright &y& Elsevier]

Abstract: In this paper we first generalize the Ostrowski inequality on time scales for k points and then unify corresponding continuous and discrete versions. We also point out some particular Ostrowski type inequalities on time scales as special cases. [Copyright &y& Elsevier]

Abstract: In this paper, a multi-scale element-free Galerkin method is presented for the Stokes problem. The new method is based on the Hughes’ variational multi-scale formulation, and arises from a decomposition of the velocity field into coarse/resolved scales and fine/unresolved scales. In this method, an unresolved model is obtained in which unresolved scales are incorporated analytically through the bubble functions. Modeling of the unresolved scales corrects the lack of the stability of the standard element-free Galerkin formulation and the resulting stabilized formulation possesses superior properties like that of the streamline upwind/Petrov–Galerkin (SUPG) method and the Galerkin/least-squares (GLS) method. The method allows equal order basis for pressure and velocity because it violates the celebrated Babuska–Brezzi condition. A significant feature of the present method is that the structure of the stabilization tensor τ appears naturally via the solution of the fine-scale problem. Numerical results for example problems confirm that this method has some excellent properties, such as better stability and accuracy. [Copyright &y& Elsevier]

Abstract: A discontinuous Galerkin method is analyzed to approximate the nonlinear Laplacian model problem. The salient feature of the proposed scheme is that it is endowed with a discrete variational principle. The convergence of the discrete approximations to the exact solution is proven. To cite this article: E. Burman, A. Ern, C. R. Acad. Sci. Paris, Ser. I 346 (2008). [Copyright &y& Elsevier]

Abstract: The interpolation step of Guruswami and Sudan’s list decoding of Reed–Solomon codes poses the problem of finding the minimal polynomial of an ideal with respect to a certain monomial order. An efficient algorithm that solves the problem is presented based on the theory of Gröbner bases of modules. In a special case, this algorithm reduces to a simple Berlekamp–Massey-like decoding algorithm. [Copyright &y& Elsevier]

Abstract: The recently proposed simulation framework of interface relaxation for developing multi-domain multi-physics simulation engines is considered. An experimental study of the behavior of two representative interface relaxation methods is presented. Three linear and one non-linear elliptic two-dimensional PDE problems are considered and they are coupled with both cartesian and general decompositions. The characteristics and the effectiveness of the proposed collaborative PDE solving framework in general, and of the two interface relaxation methods in particular are shown. [Copyright &y& Elsevier]

Abstract: A linear algebra proof is given of the fact that the nullspace of a finite-rank linear projector, on polynomials in two complex variables, is an ideal if and only if the projector is the bounded pointwise limit of Lagrange projectors, i.e., projectors whose nullspace is a radical ideal, i.e., the set of all polynomials that vanish on a certain given finite set. A characterization of such projectors is also given in the real case. More generally, a characterization is given of those finite-rank linear projectors, on polynomials in d complex variables, with nullspace an ideal that are the bounded pointwise limit of Lagrange projectors. The characterization is in terms of a certain sequence of d commuting linear maps and so focuses attention on the algebra generated by such sequences. [Copyright &y& Elsevier]

Abstract: We address the problem of determining the viewpoint of an image without referencing to or estimating explicitly the 3-D structure pictured in the image. Used for reference are instead a number of sample snapshots of the scene, each supplied with the associated viewpoint. By viewing image and its associated viewpoint as the input and output of a function, and the reference snapshot–viewpoint pairs as input–output samples of that function, we have a natural formulation of the problem as an interpolation one. The interpolation formulation allows imaging details like camera intrinsic parameters to be unknown, and the specification of the desired viewpoint to be not necessarily in metric terms. We describe an interpolation-based mechanism that determines the viewpoint of any given input image, which has the property that it fits all the given input–output reference samples exactly. Experimental results on benchmarking image datasets show that the mechanism is effective in reaching quality viewpoint solution even with only a few reference snapshots. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the linearly viscoelastic equations for Koiter shells. The existence and uniqueness of the solution are proved by Galerkin method. [Copyright &y& Elsevier]

Abstract: An original model and code for 3-D radiation of anisotropically scattering gray media is developed where radiative transfer equation (RTE) is solved by finite volume method (FVM) and scattering phase function (SPF) is defined by Mie Equations (ME). To the authors’ best knowledge this methodology was not developed before. Missing the benchmark, another new 3-D model and code, which solve the same problems, based on a combination of zone method (ZM) and Monte Carlo method (MC), as a solution of RTE, is developed. Here SPF is also calculated by Mie Equations. The conception ZM+MC is numerically expensive and is used and recommended only as a benchmark. The 3-D rectangular enclosure and the spherical geometry of particles are considered. The both models are applied: (i) to an isotropic and to four anisotropic scattering cases previously used in literature for 2-D cases and (ii) to solid particles of several various coals and of a fly ash. The agreement between the predictions obtained by these two different numerical methods for coals and ash is very good. The effects of scattering albedo and of wall reflectivity on the radiative heat flux are presented. It was found that the developed 3-D model, where FVM was coupled with ME, is reliable and accurate. The methodology is also suitable for extension towards: (i) mixture of non-gray gases with particles and (ii) incorporation in computational fluid dynamics. [Copyright &y& Elsevier]

Abstract: MCNP has stood so far as one of the main Monte Carlo radiation transport codes. Its use, as any other Monte Carlo based code, has increased as computers perform calculations faster and become more affordable along time. However, the use of Monte Carlo method to tally events in volumes which represent a small fraction of the whole system may turn to be unfeasible, if a straight analogue transport procedure (no use of variance reduction techniques) is employed and precise results are demanded. Calculations of reaction rates in activation foils placed in critical systems turn to be one of the mentioned cases. The present work takes advantage of the fixed source representation from MCNP to perform the above mentioned task in a more effective sampling way (characterizing neutron population in the vicinity of the tallying region and using it in a geometric reduced coupled simulation). An extended analysis of source dependent parameters is studied in order to understand their influence on simulation performance and on validity of results. Although discrepant results have been observed for small enveloping regions, the procedure presents itself as very efficient, giving adequate and precise results in shorter times than the standard analogue procedure. [Copyright &y& Elsevier]

Abstract: Discontinuous Galerkin methods (DG) have particular appeal in problems involving high-order derivatives since they provide a means of weakly enforcing the continuity of the unknown-field derivatives. This paper proposes a new discontinuous Galerkin method for Kirchhoff–Love shells considering only the membrane and bending response. The proposed one-field method utilizes the weak enforcement in such a way that the displacements are the only unknowns, while the rotation continuity is weakly enforced. This work presents the formulation of the new discontinuous Galerkin method for linear elastic shells, demonstrates the consistency and stability of the proposed framework, and establishes the method’s convergence rate. After a description of the formulation implementation into a finite-element code, these properties are demonstrated on numerical applications. [Copyright &y& Elsevier]

Abstract: A priori error estimates for the Rosenau equation, which is a K-dV like Rosenau equation modelled to describe the dynamics of dense discrete systems, have been studied by one of the authors. But since a priori error bounds contain the unknown solution and its derivatives, it is not effective to control error bounds with only a given step size. Thus we need to estimate a posteriori errors in order to control accuracy of approximate solutions using variable step sizes. A posteriori error estimates of the Rosenau equation are obtained by a discontinuous Galerkin method and the stability analysis is discussed for the dual problem. Numerical results on a posteriori error and wave propagation are given, which are obtained by using various spatial and temporal meshes controlled automatically by a posteriori error. [Copyright &y& Elsevier]

Abstract: The properties of fiber reinforced metal matrix composites (MMCs) are anisotropic. These properties can be more appropriately evaluated in polar coordinate. In this paper, the elastic–plastic mechanics model of MMCs is presented based on the combination of macroscopic and meso-scopic analysis in polar coordinate. The influence of micro-parameters such as matrix materials, the fibers and fiber volume fraction of Al/Mg metal matrix composites (MMCs) on the tensile properties of MMCs is also investigated numerically by the representative volume element (RVE) and the numerical results are compared with the experimental data. The results show that the model built in polar coordinate is suitable to the real constructs of MMCs; the fiber plays a major role of load supporting during MMCs stretched. And there is little distinction of the value of maximum stress of MMCs between tests and the numerical analysis. [Copyright &y& Elsevier]

Abstract: We discuss the numerical differentiation from the noisy data. We propose a method, i.e., adding a differential term, for the differentiation problem. An error estimate shows that the method allows for the approximate recovery of the derivative function. Some interesting numerical tests display the properties of the proposed method. [Copyright &y& Elsevier]

Abstract: In immersed interface methods, solids in a fluid are represented by singular forces in the Navier–Stokes equations, and flow jump conditions induced by the singular forces directly enter into numerical schemes. This paper focuses on the implementation of an immersed interface method for simulating fluid–solid interaction in 3D. The method employs the MAC scheme for the spatial discretization, the RK4 scheme for the time integration, and an FFT-based Poisson solver for the pressure Poisson equation. A fluid–solid interface is tracked by Lagrangian markers. Intersections of the interface with MAC grid lines identify finite difference stencils on which jump contributions to finite difference schemes are needed. To find the intersections and to interpolate jump conditions from the Lagrangian markers to the intersections, parametric triangulation of the interface is used. The velocity of the Lagrangian markers is interpolated directly from surrounding MAC grid nodes with interpolation schemes accounting for jump conditions. Numerical examples demonstrate that (1) the method has near second-order accuracy in the infinity norm for velocity, and the accuracy for pressure is between first and second order; (2) the method conserves the volume enclosed by a no-penetration boundary; and (3) the method can efficiently handle multiple moving solids with ease. [Copyright &y& Elsevier]

Abstract: The problem of computing bilinear Diffie–Hellman maps is considered. It is shown that the problem of computing the map is equivalent to computing a diagonal version of it. Various lower bounds on the degree of any polynomial that interpolates this diagonal version of the map are found that shows that such an interpolation will involve a polynomial of large degree, relative to the size of the set on which it interpolates. [Copyright &y& Elsevier]

Abstract: An evanescent wave is produced when a propagating incident wave impinges on an interface between two media or materials at a sub-critical angle. The sub-scale nature of such a wave makes it difficult to be captured computationally. In this paper, the Discontinuous Enrichment Method (DEM) developed in [C. Farhat, I. Harari, L.P. Franca, The discontinuous enrichment method, Comput. Methods Appl. Mech. Engrg. 190 (2001) 6455–6479; C. Farhat, I. Harari, U. Hetmaniuk, A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime, Comput. Methods Appl. Mech. Engrg. 192 (2003) 1389–1419; C. Farhat, I. Harari, U. Hetmaniuk, The discontinuous enrichment method for multiscale analysis, Comput. Methods Appl. Mech. Engrg. 192 (2003) 3195–3210; C. Farhat, R. Tezaur, P. Weidemann-Goiran, Higher-order extensions of a discontinuous Galerkin method for mid-frequency Helmholtz problems, Int. J. Numer. Methods Engrg. 61 (2004) 1938–1956; C. Farhat, P. Wiedemann-Goiran, R. Tezaur, A discontinuous Galerkin method with plane waves and Lagrange multipliers for the solution of short wave exterior Helmholtz problems on unstructured meshes. New computational methods for wave propagation, Wave Motion 39(4) (2004) 307–317; L. Zhang, R. Tezaur, C. Farhat, The discontinuous enrichment method for elastic wave propagation in the medium-frequency regime, Int. J. Numer. Methods Engrg. 66 (2006) 2086–2114; R. Tezaur, C. Farhat, Three-dimensional discontinuous Galerkin elements with plane waves and Lagrange multipliers for the solution of mid-frequency Helmholtz problems, Int. J. Numer. Methods Engrg. 66 (2006) 796–815] is extended to a class of evanescent wave problems. New DEM elements are constructed by enriching polynomial approximations with free-space evanescent solutions in order to achieve high accuracy for problems with fluid/fluid or fluid/solid interfaces on which evanescent waves may occur. The new DEM elements are applied to the solution of various two-dimensional model problems with evanescent waves. Their performance is observed to be better than that of the basic Helmholtz type DEM elements, and superior to that of the classical higher-order polynomial finite element method. [Copyright &y& Elsevier]

The article presents research on the boundary blow-up solutions to p(x)-Laplacian equations with exponential nonlinearities. It discusses the singularity of boundary blow-up solutions as well as their existence. It focuses on the differential equations and variational problems with nonstandard p(x)-growth conditions. The study aims to give the existence and asymptotic behavior of solutions for problem (P). Study shows that the space becomes a Banach space called a generalized Lebesgue space. It notes that the space is separable, reflexive and uniform convex.

Abstract: With the beginning of the new century, the European EGS Project got into its decisive state by reaching the final reservoir depth of 5km. The three boreholes, GPK2, GPK3 and GPK4 have been successfully targeted at their predefined reservoir positions. Improvement of the reservoir conditions by stimulation with a minimized seismic risk represents now a primary challenge to enable economic operation and future extension. In this context, the new HEX-S code has been developed to simulate the transient hydro-mechanical response of the rock matrix to massive hydraulic injections. The present paper describes the successful forecast of the pressure response and shearing locations for the GPK4 stimulation in September 2004. As basis for this predictive modeling, the reservoir model was derived from data analysis of the stimulation of GPK3 in May 2003. Stimulation flow rates up to 45l/s at GPK4 and of >60l/s at GPK3 have been applied, triggering several 10,000 of microseismic events. The transient numerical simulations with the HEX-S code match the main characteristics of both, the microseismic and the hydraulic behavior. Different model calculations demonstrate the capabilities of our new approach. It is noteworthy that the modeling became possible only due to the excellent data quality at the Soultz project. The results demonstrate that simulations based on solid physical ground can reveal the complex reservoir behavior during hydraulic stimulation. The use of HEX-S also provides perspectives for future developments such as design calculations that enable optimizing cost-intensive hydraulic stimulations before hand. [Copyright &y& Elsevier]

Abstract: A transient natural convection develops along vertical plane Cu electrodes during the galvanostatic electrolysis. The time variations of both the concentration profile of CuSO4 and the velocity profile of a 0.6M CuSO4 aqueous electrolyte were measured. The mass transfer rate of Cu2+ ion as well as the natural convective fluid motion was analyzed. The effect of electrode spacing on the transient natural convection was focused. The overshooting phenomenon caused by the transient diffusive and convective mass transfer rate was observed. Numerical simulation may provide as better understanding of the phenomena. [Copyright &y& Elsevier]

Abstract: This paper shows that for any suitably smooth function and arbitrarily selected interpolation nodes in , the Filon method and the Levin method for with the polynomial interpolation approach are identical when is a linear function. Based on this result, a new efficient Levin quadrature for is presented. [Copyright &y& Elsevier]