相关论文: Zeta Correction: A New Approach to Constructing Co…
In this paper we propose and analyze a general arbitrarily high-order modified trapezoidal rule for a class of weakly singular integrals of the forms $I = \int_{\mathbb{R}^n}\phi(x)s(x)dx$ in $n$ dimensions, where $\phi\in…
Symmetric polynomial quadrature rules for triangles are commonly used to efficiently integrate two-dimensional domains in finite-element-type problems. While the development of such rules focuses on the maximum degree a given number of…
This paper is a summary of the general approach outlined in my previous papers toward proving the riemann hypothesis. Numerical and graphical proof of the Riemann Hypothesis is presented with analytical arguments although more work needs…
Integral equation methods for the solution of partial differential equations, when coupled with suitable fast algorithms, yield geometrically flexible, asymptotically optimal and well-conditioned schemes in either interior or exterior…
Based upon the fast computation of the coefficients of the interpolation polynomials at Chebyshev-type points by FFT, DCT and IDST, respectively, together with the efficient evaluation of the modified moments by forwards recursions or by…
We present highlights of computations of the Riemann zeta function around large values and high zeros. The main new ingredient in these computations is an implementation of the second author's fast algorithm for numerically evaluating…
This paper is the direct-formulation companion to [Burbano-Gallegos, P\'erez-Arancibia, and Turc, ESAIM: M2AN, 60(1):273--315, 2026], which developed indirect combined-field-only boundary integral equations (BIEs) for time-harmonic…
Multidimensional diagonal-norm summation-by-parts (SBP) operators with collocated volume and facet nodes, known as diagonal-$ \mathsf{E} $ operators, are attractive for entropy-stable discretizations from an efficiency standpoint. However,…
We develop quadrature rules for the isogeometric analysis of wave propagation and structural vibrations that minimize the discrete dispersion error of the approximation. The rules are optimal in the sense that they only require two…
The isogeometric formulation of Boundary Element Method (BEM) is investigated within the adaptivity framework. Suitable weighted quadrature rules to evaluate integrals appearing in the Galerkin BEM formulation of 2D Laplace model problems…
The method of regularized stokeslets is a powerful numerical method to solve the Stokes flow equations for problems in biological fluid mechanics. A recent variation of this method incorporates a nearest-neighbor discretization to improve…
This paper introduces optimally-blended quadrature rules for isogeometric analysis and analyzes the numerical dispersion of the resulting discretizations. To quantify the approximation errors when we modify the inner products, we generalize…
Consider the scattering of a time-harmonic plane wave by a rigid obstacle embedded in a homogeneous and isotropic elastic medium in two dimensions. In this paper, a novel boundary integral formulation is proposed and its highly accurate…
A quadrature formula is a formula computing a definite integration by evaluation at finite points. The existence of certain quadrature formulas for orthogonal polynomials is related to interesting problems such as Waring's problem in number…
We present an accurate quadrature formula on the sphere with less localized quadrature points for efficient numerical computation of the radiative transport equation (RTE) in the three dimensions. High accuracy of the present method…
The sparse approximation of high-frequency Helmholtz-type integral operators has many important physical applications such as problems in wave propagation and wave scattering. The discrete system matrices are huge and densely populated;…
We employ a multivariate extension of the Gauss quadrature formula, originally due to Berens, Schmid and Xu [BSX95], so as to derive cubature rules for the integration of symmetric functions over hypercubes (or infinite limiting…
This document is made up of two different units. One of them is a regular terse research article, whereas the other one is the detailed and independently written explanations for the paper, so that readers of the short paper do not need to…
In the realm of numerical analysis, the study of structured backward errors (BEs) in saddle point problems (SPPs) has shown promising potential for development. However, these investigations overlook the inherent sparsity pattern of the…
We propose a high-precision numerical quadrature framework based on local Fourier extension (LFE) approximations. The method constructs, on each subinterval, a truncated-SVD stabilized local Fourier continuation of the integrand on an…