Interpolation of Hardy spaces on circular domains

In the paper, abstract interpolation of Hardy spaces on circular domains is studied. In particular, an extension of Peter Jones's result on the interpolation of Hardy spaces on the disc to multiply‐connected domains is presented. Moreover, some applications in approximation theory are given.     

Conservative semi‐Lagrangian finite volume schemes

Semi‐Lagrangian finite volume schemes for the numerical approximation of linear advection equations are presented. These schemes are constructed so that the conservation properties are preserved by the numerical approximation. This is achieved using an interpolation procedure based on area‐weighting. Numerical results are presented illustrating some of the features of these schemes. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:403–425, 2001     

Multiresolution schemes for strongly degenerate parabolic equations in one space dimension

An adaptive finite volume method for one‐dimensional strongly degenerate parabolic equations is presented. Using an explicit conservative numerical scheme with a third‐order Runge‐Kutta method for the time discretization, a third‐order ENO interpolation for the convective term, and adding a conservative discretization for the diffusive term, we apply the multiresolution method combining two fundamental concepts: the switch between central interpolation or exact computing of numerical flux and a thresholded wavelet transform applied to cell averages of the solution to control the switch. Applications to mathematical models of sedimentation‐consolidation processes and traffic flow with driver reaction, which involve different types of boundary conditions, illustrate the computational efficiency of the new method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Galerkin's finite element formulation of the system of fourth‐order boundary‐value problems

In this article, a Galerkin's finite element approach based on weighted‐residual is presented to find approximate solutions of a system of fourth‐order boundary‐value problems associated with obstacle, unilateral and contact problems. The approach utilizes a piece‐wise cubic approximations utilizing cubic Hermite interpolation polynomials. Numerical studies have shown the superior accuracy and lesser computational cost of the scheme in comparison to cubic spline, non‐polynomial spline and cubic non‐polynomial spline methods. Numerical examples are presented to illustrate the applicability of the method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1551–1560, 2011     

A high‐order finite difference discretization strategy based on extrapolation for convection diffusion equations

We propose a new high‐order finite difference discretization strategy, which is based on the Richardson extrapolation technique and an operator interpolation scheme, to solve convection diffusion equations. For a particular implementation, we solve a fine grid equation and a coarse grid equation by using a fourth‐order compact difference scheme. Then we combine the two approximate solutions and use the Richardson extrapolation to compute a sixth‐order accuracy coarse grid solution. A sixth‐order accuracy fine grid solution is obtained by interpolating the sixth‐order coarse grid solution using an operator interpolation scheme. Numerical results are presented to demonstrate the accuracy and efficacy of the proposed finite difference discretization strategy, compared to the sixth‐order combined compact difference (CCD) scheme, and the standard fourth‐order compact difference (FOC) scheme. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 18–32, 2004.     

Conservative Velocity Interpolation for PDF Methods

In many particle methods the accurate interpolation of a velocity field represented on a computational grid to arbitrary positions is crucial [2, 5]. Here, the importance of mass conservation and order of the interpolation scheme were analyzed. Initially equally distributed particles were tracked in a stationary, incompressible 2d flow field using different interpolation schemes. It could be demonstrated that especially mass conservation is of great importance, in particular in the case of complex flow patterns. The ideas presented in this paper are more general and the methods can be extended for unsteady, compressible 3d flow problems. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonoverlapping domain decomposition characteristic finite differences for three‐dimensional convection‐diffusion equations

One domain decomposition method modified with characteristic differences is presented for non‐periodic three‐dimensional equations by multiply‐type quadratic interpolation and variant time‐step technique. This method consists of reduced‐scale, two‐dimensional computation on subdomain interface boundaries and fully implicit subdomain computation in parallel. A computational algorithm is outlined and an error estimate in discrete l²‐ norm is established by introducing new inner products and norms. Finally, numerical examples are given to illustrate the theoretical results, efficiency and parallelism of this method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 17‐37, 2012     

Stability and convergence of spectral radial point interpolation method locally applied on two‐dimensional pseudoparabolic equation

In this article, we study a spectral meshless radial point interpolation of pseudoparabolic equations in two spatial dimensions. Shape functions, which are constructed through point interpolation method using the radial basis functions, help us to treat problem locally with the aim of high‐order convergence rate. The time derivatives are approximated by the finite difference time‐stepping method. The stability and convergence of this meshless approach are discussed and theoretically proven. Numerical results are presented to illustrate the theoretical findings. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 724–741, 2017     

Solving hyperbolic conservation laws using multiquadric quasi‐interpolation

In this article, we apply the univariate multiquadric (MQ) quasi‐interpolation to solve the hyperbolic conservation laws. At first we construct the MQ quasi‐interpolation corresponding to periodic and inflow‐outflow boundary conditions respectively. Next we obtain the numerical schemes to solve the partial differential equations, by using the derivative of the quasi‐interpolation to approximate the spatial derivative of the differential equation and a low‐order explicit difference to approximate the temporal derivative of the differential equation. Then we verify our scheme for the one‐dimensional Burgers' equation (without viscosity). We can see that the numerical results are very close to the exact solution and the computational accuracy of the scheme is ??(τ), where τ is the temporal step. We can improve the accuracy by using the high‐order quasi‐interpolation. Moreover the methods can be generalized to the other equations. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Análise de vigas laminadas utilizando o natural neighbour radial point interpolation method

In this work, a meshless method, “natural neighbour radial point interpolation method” (NNRPIM), is applied to the one‐dimensional analysis of laminated beams, considering the theory of Timoshenko.The NNRPIM combines the mathematical concept of natural neighbours with the radial point interpolation. Voronoï diagrams allows to impose the nodal connectivity and the construction of a background mesh for integration purposes, via influence cells. The construction of the NNRPIM interpolation functions is shown, and, for this, it is used the multiquadratic radial basis function. The generated interpolation functions possess infinite continuity and the delta Kronecker property, which facilitates the enforcement of boundary conditions, since these can be directly imposed, as in the finite element method (FEM).In order to obtain the displacements and the deformation fields, it is considered the Timoshenko theory for beams under transverse efforts. Several numerical examples of isotropic beams and laminated beams are presented in order to demonstrate the convergence and accuracy of the proposed application. The results obtained are compared with analytical solutions available in the literature.     

Error estimations for some meshless boundary interpolation methods

Meshless methods have become quite popular in numerical treatment of partial differential equations because of their simplicity and the fact that they require neither domain nor boundary mesh. In general, however, they convert the original problem to a highly ill‐conditioned linear system of algebraic equations with a dense matrix. Recently, a special technique has been proposed which circumvents this computational difficulty. This method, called Direct Multi‐Elliptic Interpolation Method, is based on a scattered data interpolation which defines the interpolation function as a solution of a higher order multi‐elliptic equation. Here the boundary version of this meshless method which is based on a multi‐elliptic boundary interpolation is considered. Error estimations are derived justifying the interpolation function to be a good approximation of the solution of the original boundary value problem as well. At the same time, the problem of large, dense and ill‐conditioned matrices as well as the mesh generation are completely avoided. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lagrange interpolation and finite element superconvergence

We consider the finite element approximation of the Laplacian operator with the homogeneous Dirichlet boundary condition, and study the corresponding Lagrange interpolation in the context of finite element superconvergence. For d‐dimensional Q_k‐type elements with d ≥ 1 and k ≥ 1, we prove that the interpolation points must be the Lobatto points if the Lagrange interpolation and the finite element solution are superclose in H¹ norm. For d‐dimensional P_k‐type elements, we consider the standard Lagrange interpolation—the Lagrange interpolation with interpolation points being the principle lattice points of simplicial elements. We prove for d ≥ 2 and k ≥ d + 1 that such interpolation and the finite element solution are not superclose in both H¹ and L² norms and that not all such interpolation points are superconvergence points for the finite element approximation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 33–59, 2004.     

Scattered data quasi‐interpolation on spheres

This paper studies the construction and approximation of quasi‐interpolation for spherical scattered data. First of all, a kind of quasi‐interpolation operator with Gaussian kernel is constructed to approximate the spherical function, and two Jackson type theorems are established. Second, the classical Shepard operator is extended from Euclidean space to the unit sphere, and the error of approximation by the spherical Shepard operator is estimated. Finally, the compact supported kernel is used to construct quasi‐interpolation operator for fitting spherical scattered data, where the spherical modulus of continuity and separation distance of scattered sampling points are employed as the measurements of approximation error. Copyright © 2014 John Wiley & Sons, Ltd.     

Lagrange‐type basis for multivariate uniform integrable tensor‐product Birkhoff interpolation

Birkhoff interpolation is the most general interpolation scheme. We study the Lagrange‐type basis for uniform integrable tensor‐product Birkhoff interpolation. We prove that the Lagrange‐type basis of multivariate uniform tensor‐product Birkhoff interpolation can be obtained by multiplying corresponding univariate Lagrange‐type basis when the integrable condition is satisfied. This leads to less computational complexity, which drops to from .     

An efficient pseudo‐spectral Legendre–Galerkin method for solving a nonlinear partial integro‐differential equation arising in population dynamics

The pseudo‐spectral Legendre–Galerkin method (PS‐LGM) is applied to solve a nonlinear partial integro‐differential equation arising in population dynamics. This equation is a competition model in which similar individuals are competing for the same resources. It is a kind of reaction–diffusion equation with integral term corresponding to nonlocal consumption of resources. The proposed method is based on the Legendre–Galerkin formulation for the linear terms and interpolation operator at the Chebyshev–Gauss–Lobatto (CGL) points for the nonlinear terms. Also, the integral term, which is a kind of convolution, is directly computed by a fast and accurate method based on CGL interpolation operator, and thus, the use of any quadrature formula in its computation is avoided. The main difference of the PS‐LGM presented in the current paper with the classic LGM is in treating the nonlinear terms and imposing boundary conditions. Indeed, in the PS‐LGM, the nonlinear terms are efficiently handled using the CGL points, and also the boundary conditions are imposed strongly as collocation methods. Combination of the PS‐LGM with a semi‐implicit time integration method such as second‐order backward differentiation formula and Adams‐Bashforth method leads to reducing the complexity of computations and obtaining a linear algebraic system of equations with banded coefficient matrix. The desired equation is considered on one and two‐dimensional spatial domains. Efficiency, accuracy, and convergence of the proposed method are demonstrated numerically in both cases. Copyright © 2012 John Wiley & Sons, Ltd.     

Notes on ferromagnetic p‐spin and REM

In this paper we apply some of the recent mathematical techniques (mainly based on interpolation) developed in the spin glass theory to the ferromagnetic p‐spin model. We introduce two Hamiltonians and derive their thermodynamics. This is a second step toward an alternative and rigorous formulation of the statistical mechanics of simple systems on lattice. A first step has been performed in J. Stat. Phys. (2007; arXiv:0712.1344) where the techniques have been tested on the two‐body Ising model. For completeness the adaptation of the well‐known random energy model to the context of the ferromagnetism is presented. At the end a discussion on the extension of these techniques to Gaussian‐disordered p‐spin models is also briefly outlined. Copyright © 2008 John Wiley & Sons, Ltd.     

Spectral meshless radial point interpolation (SMRPI) method to two‐dimensional fractional telegraph equation

H. Ammari In this article, an innovative technique so‐called spectral meshless radial point interpolation (SMRPI) method is proposed and, as a test problem, is applied to a classical type of two‐dimensional time‐fractional telegraph equation defined by Caputo sense for (1 < α≤2). This new methods is based on meshless methods and benefits from spectral collocation ideas, but it does not belong to traditional meshless collocation methods. The point interpolation method with the help of radial basis functions is used to construct shape functions, which play as basis functions in the frame of SMRPI method. These basis functions have Kronecker delta function property. Evaluation of high‐order derivatives is not difficult by constructing operational matrices. In SMRPI method, it does not require any kind of integration locally or globally over small quadrature domains, which is essential of the finite element method (FEM) and those meshless methods based on Galerkin weak form. Also, it is not needed to determine strict value for the shape parameter, which plays an important role in collocation method based on the radial basis functions (Kansa's method). Therefore, computational costs of SMRPI method are less expensive. Two numerical examples are presented to show that SMRPI method has reliable rates of convergence. Copyright © 2015 John Wiley & Sons, Ltd.     

A Hybrid Variational Model for the Analysis of Free‐Corner Effects in Layered Plates

A variational model for the assessment of free‐edge and free‐corner effects [1‐3] in thermally loaded rectangular cross‐ply laminate plates is presented. The physical layers of the plate are discretized by an arbitrary number of mathematical layers through the laminate thickness. A layerwise displacement field with unknown interface functions which depend on the inplane coordinates is formulated wherein an a priori assumed layerwise linear thickness interpolation scheme is employed. The application of the principle of minimum elastic potential yields a set of governing Euler‐Lagrange differential equations for the unknown inplane functions which for the special free‐corner problem in rectangular cross‐ply plates can be solved in a closed‐form analytical manner. Boundary conditions of traction free laminate edges are fulfilled in an average sense. Since the approach utilizes a discretization through the plate thickness yet allows for closed‐form solutions within the layer planes for all state variables it is appropriate to speak of a hybrid analysis approach. The presented method allows easy application, can be run on every standard personal computer and is in favourable agreement with comparative finite element calculations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new multiquadric quasi‐interpolation operator with interpolation property

In this article, we discuss a class of multiquadric quasi‐interpolation operator that is primarily on the basis of Wu–Schaback's quasi‐interpolation operator and radial basis function interpolation. The proposed operator possesses the advantages of linear polynomial reproducing property, interpolation property, and high accuracy. It can be applied to construct flexible function approximation and scattered data fitting from numerical experiments. Copyright © 2013 John Wiley & Sons, Ltd.