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In this paper we analyze a mixed displacement‐pseudostress formulation for the elasticity eigenvalue problem. We propose a finite element method to approximate the pseudostress tensor with Raviart–Thomas elements and the displacement with piecewise polynomials. With the aid of the classic theory for compact operators, we prove that our method is convergent and does not introduce spurious modes. Error estimates for the proposed method are derived. Finally, we report some numerical tests supporting the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the existence of solutions for a nonlocal Sturm–Liouville problem with composite fractional derivatives under some initial value conditions. Furthermore, applying above results and operator theory, we specifically research the geometric multiplicity of eigenvalues for the nonlocal Sturm–Liouville eigenvalue problem. [ABSTRACT FROM AUTHOR]

Summary: The FEAST eigenvalue algorithm is a subspace iteration algorithm that uses contour integration to obtain the eigenvectors of a matrix for the eigenvalues that are located in any user‐defined region in the complex plane. By computing small numbers of eigenvalues in specific regions of the complex plane, FEAST is able to naturally parallelize the solution of eigenvalue problems by solving for multiple eigenpairs simultaneously. The traditional FEAST algorithm is implemented by directly solving collections of shifted linear systems of equations; in this paper, we describe a variation of the FEAST algorithm that uses iterative Krylov subspace algorithms for solving the shifted linear systems inexactly. We show that this iterative FEAST algorithm (which we call IFEAST) is mathematically equivalent to a block Krylov subspace method for solving eigenvalue problems. By using Krylov subspaces indirectly through solving shifted linear systems, rather than directly using them in projecting the eigenvalue problem, it becomes possible to use IFEAST to solve eigenvalue problems using very large dimension Krylov subspaces without ever having to store a basis for those subspaces. IFEAST thus combines the flexibility and power of Krylov methods, requiring only matrix–vector multiplication for solving eigenvalue problems, with the natural parallelism of the traditional FEAST algorithm. We discuss the relationship between IFEAST and more traditional Krylov methods and provide numerical examples illustrating its behavior. [ABSTRACT FROM AUTHOR]

The Adaptive Finite Element Method (AFEM) for approximating solutions of PDE boundary value and eigenvalue problems is a numerical scheme that automatically and iteratively adapts the finite element space until a sufficiently accurate approximate solution is found. The adaptation process is based on a posteriori error estimators, and at each step of this process an algebraic problem (linear or nonlinear algebraic system or eigenvalue problem) has to be solved. In practical computations the solution of the algebraic problem cannot be obtained exactly. As a consequence, the algebraic error should be incorporated in the context of the AFEM and its a posteriori error estimators. The goal of this paper is to survey some existing approaches in the AFEM context that consider the interplay between the finite element discretization and the algebraic computation. We believe that a better understanding of this interplay is of great importance for the future development in the area of numerically solving large-scale real-world motivated problems. (© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

Numerical solution of nonlinear eigenvalue problems (NEPs) is frequently encountered in computational science and engineering. The applicability of most existing methods is limited by the matrix structures, properties of the eigen-solutions, sizes of the problems, etc. This paper aims to remove those limitations and develop robust and universal NEP solvers for large-scale engineering applications. The novelty lies in two aspects. First, a rational interpolation approach (RIA) is proposed based on the Keldysh theorem for holomorphic matrix functions. Comparing with the existing contour integral approach, the RIA provides the possibility to select sampling points in more general regions and has advantages in improving the accuracy and reducing the computational cost. Second, a resolvent sampling scheme using the RIA is proposed to construct reliable search spaces for the Rayleigh-Ritz procedure, based on which a robust eigen-solver, called resolvent sampling based Rayleigh-Ritz method (RSRR), is developed for solving general NEPs. The RSRR can be easily implemented and parallelized. The advantages of the RIA and the performance of the RSRR are demonstrated by a variety of benchmark and application examples. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, an efficient local radial basis function collocation method (LRBFCM) is presented for computing the band structures of the two-dimensional (2D) solid/fluid and fluid/solid phononic crystals. Both systems of solid scatterers embedded in a fluid matrix (solid/fluid phononic crystals) and fluid scatterers embedded in a solid matrix (fluid/solid phononic crystals) are investigated. The solid-fluid interactions are taken into account by properly formulating and treating the continuity/equilibrium conditions on the solid-fluid interfaces, which require an accurate computation of the normal derivatives of the displacements and the pressure on the fluid-solid interfaces and the unit-cell boundaries. The developed LRBFCM for the mixed wave propagation problems in 2D solid/fluid and fluid/solid phononic crystals is validated by the corresponding results obtained by the finite element method (FEM). To the best knowledge of the authors, the present LRBFCM has yet not been applied to the band structure computations of 2D solid/fluid and fluid/solid phononic crystals. For different lattice forms, scatterers' shapes, acoustic impedance ratios, and material combinations (solid scatterers in fluid matrix or fluid scatterers in solid matrix), numerical results are presented and discussed to reveal the efficiency and the accuracy of the developed LRBFCM for calculating the band structures of 2D solid/fluid and fluid/solid phononic crystals. A comparison of the present numerical results with that of the FEM shows that the present LRBFCM is much more efficient than the FEM for the band structure computations of the considered 2D solid/fluid and fluid/solid phononic crystals. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we study an a posteriori error indicator introduced in E. Dari, R.G. Durán, and C. Padra, Appl. Numer. Math., 2012, for the approximation of the Laplace eigenvalue problem with Crouzeix-Raviart nonconforming finite elements. In particular, we show that the estimator is robust also in presence of eigenvalues of multiplicity greater than one. Some numerical examples confirm the theory and illustrate the convergence of an adaptive algorithm when dealing with multiple eigenvalues. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, two accelerated divide-and-conquer (ADC) algorithms are proposed for the symmetric tridiagonal eigenvalue problem, which cost O(N2r) flops in the worst case, where N is the dimension of the matrix and r is a modest number depending on the distribution of eigenvalues. Both of these algorithms use hierarchically semiseparable (HSS) matrices to approximate some intermediate eigenvector matrices, which are Cauchy-like matrices and are off-diagonally low-rank. The difference of these two versions lies in using different HSS construction algorithms, one (denoted by ADC1) uses a structured low-rank approximation method and the other (ADC2) uses a randomized HSS construction algorithm. For the ADC2 algorithm, a method is proposed to estimate the off-diagonal rank. Numerous experiments have been carried out to show their stability and efficiency. These algorithms are implemented in parallel in a shared memory environment, and some parallel implementation details are included. Comparing the ADCs with highly optimized multithreaded libraries such as Intel MKL, we find that ADCs could be more than six times faster for some large matrices with few deflations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate Poincaré type inequalities for the functions having zero mean value on the whole boundary of a Lipschitz domain or on a measurable part of the boundary. We find exact and easily computable constants in these inequalities for some basic domains (rectangles, cubes, and right triangles) and discuss applications of the inequalities to quantitative analysis of partial differential equations. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper we prove a sharp upper bound for the first Dirichlet eigenvalue of a class of nonlinear elliptic operators which includes the operator [ABSTRACT FROM AUTHOR]

Recently a finite-difference frequency-domain (FDFD) formulation has been reported for the dispersion analysis of uniform waveguides loaded with anisotropic dielectrics characterized by a diagonal tensor [4]. This formulation, which leads to an eigenvalue problem for the propagation constant γ, involves four field components. This paper shows that this formulation can be improved by taking advantage of the fact that the structures considered are strictly bidirectional waveguides and thus the dispersion analysis problem can be formulated as an eigenvalue problem for γ2, involving only two field components. This improved formulation is applied to several waveguiding structures and a comparison of the results with those obtained by the commercial simulator Agilent-HFSS 5.6 show good agreement. © 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 38: 331–335, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.11052 [ABSTRACT FROM AUTHOR]

We investigate the eigenvalue problem for the Schrödinger–Poisson system as follows: −Δu+μV(x)u+K(x)ϕu=λf(x)u+g(x)|u|p−2uinℝ3,−Δϕ=K(x)u2inℝ3,$$ \left\{\begin{array}{ll}-\Delta u+\mu V(x)u+K(x)\phi u=\lambda f(x)u+g(x){\left|u\right|}^{p-2}u\kern0.20em & \mathrm{in}\kern0.51em {\mathbb{R}}^3,\\ {}-\Delta \phi =K(x){u}^2\kern0.20em & \kern0.1em \mathrm{in}\kern0.51em {\mathbb{R}}^3,\end{array}\right. $$where 20$$ {\lambda}_1\left({f}_{\Omega}\right)>0 $$ is the principal eigenvalue of −Δ$$ -\Delta $$ in H01(Ω)$$ {H}_0^1\left(\Omega \right) $$ with weight function fΩ:=f|Ω,$$ {\left.{f}_{\Omega}:= f\right|}_{\Omega}, $$ and ϕ1>0$$ {\phi}_1>0 $$ is the corresponding principal eigenfunction. Moreover, an interesting phenomenon is that when ∫Ωgϕ1pdx>0$$ {\int}_{\Omega}\kern3pt g{\phi}_1^p dx>0 $$, one positive solution is always present near λ1(fΩ)$$ {\lambda}_1\left({f}_{\Omega}\right) $$ for λ<λ1(fΩ)$$ \lambda <{\lambda}_1\left({f}_{\Omega}\right) $$ unaffected by the non‐local term, however large its norm might be. [ABSTRACT FROM AUTHOR]

A multiple monopole method based on the generalized multipole technique is presented for the calculation of band structures of two-dimensional mixed solid/fluid phononic crystals. In this method, the fields are expanded by using the fundamental solutions with multiple origins. Besides the sources used to expand the wave fields, an extra monopole source is introduced as the external excitation. By varying the frequency of the excitation, the eigenvalues can be localized as the extreme points of an appropriately chosen function. By sweeping the frequency range of interest and sweeping the boundary of the irreducible first Brillouin zone, the band structure of the phononic crystals can be obtained. The method can consider the fluid-solid interface conditions and the transverse wave mode in the solid component strictly. Some typical examples are illustrated to discuss the accuracy of the present method. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We prove that the unitary factor appearing in the QR factorization of a suitably defined rational Krylov matrix transforms a Hermitian matrix having pairwise distinct eigenvalues into a diagonal-plus-semiseparable form with prescribed diagonal term. This transformation is essentially uniquely defined by its first column. Furthermore, we prove that the set of Hermitian diagonal-plus-semiseparable matrices is invariant under QR iteration. These and other results are shown to be the rational counterpart of known facts involving structured matrices related to polynomial computations. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper the Carey non-conforming finite element is considered for solving eigenvalue problems of the second-order elliptic operator. Based on an interpolation postprocessing, high-accuracy estimates of both eigenfunctions and eigenvalues are obtained: $$\Vert \Pi ^{2}_{2h}u_{h}-u\Vert _{1}\leq Ch^{2}\Vert u\Vert _{3}$$ $$0\leq \hat{\lambda }_{h}-\lambda \leq Ch^{4}\Vert u\Vert ^{2}_{3}$$ Here, Π22h is an interpolation operator, (λ, u) and (λh, uh) are eigenpairs for the exact problem and its Carey element approximations, respectively, and $$ \hat{\lambda }_{h}$$ is defined by the Rayleigh ratio of Π22huh. Copyright © 1999 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this study, we consider a boundary value problem generated by a second‐order dynamic equation on a time scale and boundary conditions depending on the spectral parameter. We give some properties of the solutions and obtain a formulation of the number of eigenvalues of the problem. [ABSTRACT FROM AUTHOR]

Given a bounded open set [ABSTRACT FROM AUTHOR]

We address the solution of large-scale eigenvalue problems that appear in the motion simulation of complex macromolecules on multithreaded platforms, consisting of multicore processors and possibly a graphics processor (graphics processing unit). In particular, we compare specialized implementations of several high-performance eigensolvers that, by relying on disk storage and out-of-core techniques, can in principle tackle the large memory requirements of these biological problems, which in general do not fit into the main memory of current desktop machines. All these out-of-core eigensolvers, except for one, are composed of compute-bound (i.e., arithmetically intensive) operations, which we accelerate by exploiting the performance of current multicore processors and, in some cases, by additionally off-loading certain parts of the computation to a graphics processing unit accelerator. One of the eigensolvers is a memory-bound algorithm, which strongly constrains its performance when the data is on disk. However, this method exhibits a much lower arithmetic cost compared with its compute-bound alternatives for this particular application. Experimental results on a desktop platform, representative of current server technology, illustrate the potential of these methods to address the simulation of biological activity. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

During the last years, low-rank tensor approximation has been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey attempts to give a literature overview of current developments in this area, with an emphasis on function-related tensors. (© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

A parallel algorithm for reordering the eigenvalues in the real Schur form of a matrix is presented and discussed. Our novel approach adopts computational windows and delays multiple outside-window updates until each window has been completely reordered locally. By using multiple concurrent windows the parallel algorithm has a high level of concurrency, and most work is level 3 BLAS operations. The presented algorithm is also extended to the generalized real Schur form. Experimental results for ScaLAPACK-style Fortran 77 implementations on a Linux cluster confirm the efficiency and scalability of our algorithms in terms of more than 16 times of parallel speedup using 64 processors for large-scale problems. Even on a single processor our implementation is demonstrated to perform significantly better compared with the state-of-the-art serial implementation. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Numerical solving of real-world problems typically consists of several stages. After a mathematical description of the problem and its proper reformulation and discretisation, the resulting linear algebraic problem has to be solved. We focus on this last stage, and specifically consider numerical stability of iterative methods in matrix computations. In iterative methods, rounding errors have two main effects: They can delay convergence and they can limit the maximal attainable accuracy. It is important to realize that numerical stability analysis is not about derivation of error bounds or estimates. Rather the goal is to find algorithms and their parts that are safe (numerically stable), and to identify algorithms and their parts that are not. Numerical stability analysis demonstrates this important idea, which also guides this contribution. In our survey we first recall the concept of backward stability and discuss its use in numerical stability analysis of iterative methods. Using the backward error approach we then examine the surprising fact that the accuracy of a (final) computed result may be much higher than the accuracy of intermediate computed quantities. We present some examples of rounding error analysis that are fundamental to justify numerically computed results. Our points are illustrated on the Lanczos method, the conjugate gradient (CG) method and the generalised minimal residual (GMRES) method. [ABSTRACT FROM AUTHOR]

A finite-difference procedure which requires the use of virtual and duplicate nodes is presented. This procedure incorporates the Neumann and Dirichlet boundary conditions into the two-dimensional Helmholtz equation. Comparisons made with analytic solutions for waveguide modes indicate that robust solutions can be obtained with a low mesh density. © 1998 John Wiley & Sons, Inc. Microwave Opt Technol Lett 18: 86–90, 1998. [ABSTRACT FROM AUTHOR]