共查询到20条相似文献,搜索用时 531 毫秒
1.
We study the properties of an approximation of the Laplace operator with Neumann boundary conditions using volume penalization. For the one-dimensional Poisson equation we compute explicitly the exact solution of the penalized equation and quantify the penalization error. Numerical simulations using finite differences allow then to assess the discretization and penalization errors. The eigenvalue problem of the penalized Laplace operator with Neumann boundary conditions is also studied. As examples in two space dimensions, we consider a Poisson equation with Neumann boundary conditions in rectangular and circular domains. 相似文献
2.
There is a family of potentials that minimize the lowest eigenvalue of a Schrödinger operator under the constraint of a given L p norm of the potential. We give effective estimates for the amount by which the eigenvalue increases when the potential is not one of these optimal potentials. Our results are analogous to those for the isoperimetric problem and the Sobolev inequality. We also prove a stability estimate for Hölder’s inequality, which we believe to be new. 相似文献
3.
An L
2-estimate of the finite element error is proved for a Dirichlet and a Neumann boundary value problem on a three-dimensional,
prismatic and non-convex domain that is discretized by an anisotropic tetrahedral mesh. To this end, an approximation error
estimate for an interpolation operator that is preserving the Dirichlet boundary conditions is given. The challenge for the
Neumann problem is the proof of a local interpolation error estimate for functions from a weighted Sobolev space. 相似文献
4.
Bijaya Laxmi Panigrahi Gnaneshwar Nelakanti 《Applied mathematics and computation》2011,218(4):1222-1232
We consider the approximation of eigenfunctions of a compact integral operator with a smooth kernel by the Galerkin method using wavelet bases. By truncating the Galerkin operator, we obtain a sparse representation of a matrix eigenvalue problem. We prove that the error bounds for the eigenvalues and for the distance between the spectral subspaces are of the orders O(nμ-2nr) and O(μ-nr), respectively, where μ−n denotes the norm of the partition and r denotes the order of the wavelet basis functions. By iterating the eigenvectors, we show that the error bounds for the eigenvectors are of the order O(nμ-2nr). We illustrate our results with numerical results. 相似文献
5.
For a sparse non-singular matrix A, generally A~(-1)is a dense matrix. However, for a class of matrices,A~(-1)can be a matrix with off-diagonal decay properties, i.e., |A_(ij)~(-1)| decays fast to 0 with respect to the increase of a properly defined distance between i and j. Here we consider the off-diagonal decay properties of discretized Green's functions for Schr¨odinger type operators. We provide decay estimates for discretized Green's functions obtained from the finite difference discretization, and from a variant of the pseudo-spectral discretization. The asymptotic decay rate in our estimate is independent of the domain size and of the discretization parameter.We verify the decay estimate with numerical results for one-dimensional Schr¨odinger type operators. 相似文献
6.
We consider a noncoercive convection-diffusion problem with Neumann boundary conditions appearing in modeling of magnetic fluid seals. The associated operator has a non-trivial one-dimensional kernel spanned by a positive function. A discretization is proposed preserving these properties. Optimal error estimates in the H1-norm are based on a discrete stability result. Numerical results confirm the theoretical predictions. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
7.
We propose a new localization result for the leading eigenvalue and eigenvector of a symmetric matrix A. The result exploits the Frobenius inner product between A and a given rank-one landmark matrix X. Different choices for X may be used, depending on the problem under investigation. In particular, we show that the choice where X is the all-ones matrix allows to estimate the signature of the leading eigenvector of A, generalizing previous results on Perron-Frobenius properties of matrices with some negative entries. As another application we consider the problem of community detection in graphs and networks. The problem is solved by means of modularity-based spectral techniques, following the ideas pioneered by Miroslav Fiedler in mid-’70s. 相似文献
8.
It is still an open problem to prove a priori error estimates for finite volume schemes of higher order MUSCL type, including limiters, on unstructured meshes, which show some improvement compared to first order schemes. In this paper we use these higher order schemes for the discretization of convection dominated elliptic problems in a convex bounded domain Ω in R2 and we can prove such kind of an a priori error estimate. In the part of the estimate, which refers to the discretization of the convective term, we gain h1/2. Although the original problem is linear, the numerical problem becomes nonlinear, due to MUSCL type reconstruction/limiter technique. 相似文献
9.
The nonnegative inverse eigenvalue problem is that given a family of complex numbers λ={λ1,…,λn}, find a nonnegative matrix of order n with spectrum λ. This problem is difficult and remains unsolved partially. In this paper, we focus on its generalization that the reconstructed nonnegative matrices should have some prescribed entries. It is easy to see that this new problem will come back to the common nonnegative inverse eigenvalue problem if there is no constraint of the locations of entries. A numerical isospectral flow method which is developed by hybridizing the optimization theory and steepest descent method is used to study the reconstruction. Moreover, an error estimate of the numerical iteration for ordinary differential equations on the matrix manifold is presented. After that, a numerical method for the nonnegative symmetric inverse eigenvalue problem with prescribed entries and its error estimate are considered. Finally, the approaches are verified by the numerical test results. 相似文献
10.
We study the well-posedness of the Cauchy problem with Dirichlet or Neumann boundary conditions associated to an H1-critical semilinear wave equation on a smooth bounded domain Ω⊂R2. First, we prove an appropriate Strichartz type estimate using the Lq spectral projector estimates of the Laplace operator. Our proof follows Burq, Lebeau and Planchon (2008) [4]. Then, we show the global well-posedness when the energy is below or at the threshold given by the sharp Moser-Trudinger inequality. Finally, in the supercritical case, we prove an instability result using the finite speed of propagation and a quantitative study of the associated ODE with oscillatory data. 相似文献
11.
George C. Hsiao 《Journal of Computational and Applied Mathematics》2011,235(17):4949-4965
In this paper, we are concerned with the error analysis for the finite element solution of the two-dimensional exterior Neumann boundary value problem in acoustics. In particular, we establish explicit priori error estimates in H1 and L2- norms including both the effect of the truncation of the DtN mapping and that of the numerical discretization. To apply the finite element method (FEM) to the exterior problem, the original boundary value problem is reduced to an equivalent nonlocal boundary value problem via a Dirichlet-to-Neumann (DtN) mapping represented in terms of the Fourier expansion series. We discuss essential features of the corresponding variational equation and its modification due to the truncation of the DtN mapping in appropriate function spaces. Numerical tests are presented to validate our theoretical results. 相似文献
12.
A hierarchical matrix is an efficient data-sparse representation of a matrix, especially useful for large dimensional problems. It consists of low-rank subblocks leading to low memory requirements as well as inexpensive computational costs. In this work, we discuss the use of the hierarchical matrix technique in the numerical solution of a large scale eigenvalue problem arising from a finite rank discretization of an integral operator. The operator is of convolution type, it is defined through the first exponential-integral function and, hence, it is weakly singular. We develop analytical expressions for the approximate degenerate kernels and deduce error upper bounds for these approximations. Some computational results illustrating the efficiency and robustness of the approach are presented. 相似文献
13.
When the standard Chebyshev collocation method is used to solve a third order differential equation with one Neumann boundary
condition and two Dirichlet boundary conditions, the resulting differentiation matrix has spurious positive eigenvalues and
extreme eigenvalue already reaching O(N
5 for N = 64. Stable time-steps are therefore very small in this case. A matrix operator with better stability properties is obtained
by using the modified Chebyshev collocation method, introduced by Kosloff and Tal Ezer [3]. By a correct choice of mapping
and implementation of the Neumann boundary condition, the matrix operator has extreme eigenvalue less than O(N
4. The pseudospectral and modified pseudospectral methods are implemented for the solution of one-dimensional third-order partial
differential equations and the accuracy of the solutions compared with those by finite difference techniques. The comparison
verifies the stability analysis and the modified method allows larger time-steps. Moreover, to obtain the accuracy of the
pseudospectral method the finite difference methods are substantially more expensive. Also, for the small N tested, N ⩽ 16, the modified pseudospectral method cannot compete with the standard approach.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
14.
Wolf-Jürgen Beyn 《Linear algebra and its applications》2012,436(10):3839-3863
We propose a numerical method for computing all eigenvalues (and the corresponding eigenvectors) of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane. The method uses complex integrals of the resolvent operator, applied to at least k column vectors, where k is the number of eigenvalues inside the contour. The theorem of Keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension k. No initial approximations of eigenvalues and eigenvectors are needed. The method is particularly suitable for moderately large eigenvalue problems where k is much smaller than the matrix dimension. We also give an extension of the method to the case where k is larger than the matrix dimension. The quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours. Using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour. 相似文献
15.
Yury Arlinski? 《Journal of Functional Analysis》2008,254(1):154-195
The main issue we address in the present paper are the new models for completely nonunitary contractions with rank one defect operators acting on some Hilbert space of dimension N?∞. These models complement nicely the well-known models of Livšic and Sz.-Nagy-Foias. We show that each such operator acting on some finite-dimensional (respectively, separable infinite-dimensional Hilbert space) is unitarily equivalent to some finite (respectively semi-infinite) truncated CMV matrix obtained from the “full” CMV matrix by deleting the first row and the first column, and acting in CN (respectively ?2(N)). This result can be viewed as a nonunitary version of the famous characterization of unitary operators with a simple spectrum due to Cantero, Moral and Velázquez, as well as an analog for contraction operators of the result from [Yu. Arlinski?, E. Tsekanovski?, Non-self-adjoint Jacobi matrices with a rank-one imaginary part, J. Funct. Anal. 241 (2006) 383-438] concerning dissipative non-self-adjoint operators with a rank one imaginary part. It is shown that another functional model for contractions with rank one defect operators takes the form of the compression f(ζ)→PK(ζf(ζ)) on the Hilbert space L2(T,dμ) with a probability measure μ onto the subspace K=L2(T,dμ)?C. The relationship between characteristic functions of sub-matrices of the truncated CMV matrix with rank one defect operators and the corresponding Schur iterates is established. We develop direct and inverse spectral analysis for finite and semi-infinite truncated CMV matrices. In particular, we study the problem of reconstruction of such matrices from their spectrum or the mixed spectral data involving Schur parameters. It is pointed out that if the mixed spectral data contains zero eigenvalue, then no solution, unique solution or infinitely many solutions may occur in the inverse problem for truncated CMV matrices. The uniqueness theorem for recovered truncated CMV matrix from the given mixed spectral data is established. In this part the paper is closely related to the results of Hochstadt and Gesztesy-Simon obtained for finite self-adjoint Jacobi matrices. 相似文献
16.
We give a formula relating the L 2-isoperimetric profile to the spectral distribution of a Laplace operator on a finitely generated group Γ. We prove the asymptotic stability of the spectral distribution under changes of measures with finite second moment. As a consequence, we can apply techniques from geometric group theory to estimate the spectral distribution of the Laplace operator in terms of the growth and the Følner’s function of the group. This leads to upper bounds on spectral distributions of some non-solvable amenable groups and to sharp estimates of the spectral distributions of some solvable groups with exponential growth. 相似文献
17.
Lorenzo Brasco Giovanni Franzina 《NoDEA : Nonlinear Differential Equations and Applications》2013,20(6):1795-1830
We study the Stekloff eigenvalue problem for the so-called pseudo p-Laplacian operator. After proving the existence of an unbounded sequence of eigenvalues, we focus on the first nontrivial eigenvalue σ2,p , providing various equivalent characterizations for it. We also prove an upper bound for σ2,p in terms of geometric quantities. The latter can be seen as the nonlinear analogue of the Brock–Weinstock inequality for the first nontrivial Stekloff eigenvalue of the (standard) Laplacian. Such an estimate is obtained by exploiting a family of sharp weighted Wulff inequalities, which are here derived and appear to be interesting in themselves. 相似文献
18.
We consider a nonlinear Neumann problem driven by the p-Laplacian and with a Carathéodory reaction which satisfies only a unilateral growth restriction. Using the principal eigenvalue of an eigenvalue problem involving the Neumann p-Laplacian plus an indefinite potential, we produce necessary and sufficient conditions for the existence and uniqueness of positive smooth solutions. 相似文献
19.
Sergey I. Solov’ëv 《Linear algebra and its applications》2006,415(1):210-229
This paper proposes new iterative methods for the efficient computation of the smallest eigenvalue of symmetric nonlinear matrix eigenvalue problems of large order with a monotone dependence on the spectral parameter. Monotone nonlinear eigenvalue problems for differential equations have important applications in mechanics and physics. The discretization of these eigenvalue problems leads to nonlinear eigenvalue problems with very large sparse ill-conditioned matrices monotonically depending on the spectral parameter. To compute the smallest eigenvalue of large-scale matrix nonlinear eigenvalue problems, we suggest preconditioned iterative methods: preconditioned simple iteration method, preconditioned steepest descent method, and preconditioned conjugate gradient method. These methods use only matrix-vector multiplications, preconditioner-vector multiplications, linear operations with vectors, and inner products of vectors. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem. 相似文献
20.
Daisuke Koyama 《Journal of Computational and Applied Mathematics》2009,232(1):109-121
A priori error estimates in the H1- and L2-norms are established for the finite element method applied to the exterior Helmholtz problem, with modified Dirichlet-to-Neumann (MDtN) boundary condition. The error estimates include the effect of truncation of the MDtN boundary condition as well as that of discretization of the finite element method. The error estimate in the L2-norm is sharper than that obtained by the author [D. Koyama, Error estimates of the DtN finite element method for the exterior Helmholtz problem, J. Comput. Appl. Math. 200 (1) (2007) 21-31] for the truncated DtN boundary condition. 相似文献