相关论文: Accurate Spectral Collocation Computation of High …

In this paper we present a methodology for increasing the accuracy and accelerating the convergence of numerical methods for solution of Maxwell's equations in the frequency domain by taking into account the be-havior of the electromagnetic…

We present a novel integral-equation algorithm for evaluation of Zaremba eigenvalues and eigenfunctions}, that is, eigenvalues and eigenfunctions of the Laplace operator with mixed Dirichlet-Neumann boundary conditions; of course, (slight…

Stellar convection poses two main gargantuan challenges for astrophysical fluid solvers: low-Mach number flows and minuscule perturbations over steeply stratified hydrostatic equilibria. Most methods exhibit excessive numerical diffusion…

We present a novel methodology for the numerical solution of problems of diffraction by infinitely thin screens in three dimensional space. Our approach relies on new integral formulations as well as associated high-order quadrature rules.…

We initiate a systematic implementation of the spectral domain decomposition technique with the Galerkin-Collocation (GC) method in situations of interest such as the spherical collapse of a scalar field in the characteristic formulation.…

The paper analyzes and compares some spectral filtering methods as truncated singular/eigen-value decompositions and Tikhonov/Re-blurring regularizations in the case of the recently proposed Reflective [M.K. Ng, R.H. Chan, and W.C. Tang, A…

We propose a new method to solve eigenvalue problems for linear and semilinear second order differential operators in high dimensions based on deep neural networks. The eigenvalue problem is reformulated as a fixed point problem of the…

We consider an inverse spectral problem with the third-order differential equation and the non-separated boundary conditions. Two theorems on the uniqueness of the solution of this problem are proved, and a method for establishing the…

The drum problem-finding the eigenvalues and eigenfunctions of the Laplacian with Dirichlet boundary condition-has many applications, yet remains challenging for general domains when high accuracy or high frequency is needed. Boundary…

We consider the bi-Laplacian eigenvalue problem for the modes of vibration of a thin elastic plate with a discrete set of clamped points. A high-order boundary integral equation method is developed for efficient numerical determination of…

Moved by the need for rigorous and reliable numerical tools for the analysis of peridynamic materials, the authors propose a model able to capture the dispersive features of nonlocal soliton-like solutions obtained by a peridynamic…

We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schr\"odinger equation. We prove that a particular class of integrators are conjugate to unitary methods for…

A new approach is presented for the solution of spectral problems on infinite domains with regular ends, which avoids the need to solve boundary value problems for many trial values of the spectral parameter. We present numerical results…

In this paper, we consider the unique continuation problem for the Schr\"odinger equations. We prove a H\"older type conditional stability estimate and build up a parameterized stabilized finite element scheme adaptive to the \textit{a…

We study random eigenvalue problems in the context of spectral stochastic finite elements. In particular, given a parameter-dependent, symmetric positive-definite matrix operator, we explore the performance of algorithms for computing its…

The spectral renormalization method was introduced in 2005 as an effective way to compute ground states of nonlinear Schr\"odinger and Gross-Pitaevskii type equations. In this paper, we introduce an orthogonal spectral renormalization (OSR)…

In this paper, we investigate the eigenvalue problem for a non-local dispersal operator defined on a bounded spatial domain with Neumann-type boundary conditions. Unlike the classical Laplacian, the non-local operator lacks compactness,…

We study oscillations in the eigenfunctions for a fractional Schr\"odinger operator on the real line. An argument in the spirit of Courant's nodal domain theorem applies to an associated local problem in the upper half plane and provides a…

We study location of eigenvalues of one-dimensional discrete Schr\"odinger operators with complex $\ell^{p}$-potentials for $1\leq p\leq \infty$. In the case of $\ell^{1}$-potentials, the derived bound is shown to be optimal. For $p>1$, two…

A finite difference method is constructed to solve singularly perturbed convection-diffusion problems posed on smooth domains. Constraints are imposed on the data so that only regular exponential boundary layers appear in the solution. A…