The quadrature error associated with a regular quadrature rule for evaluation of a layer potential increases rapidly when the evaluation point approaches the surface and the integral becomes nearly singular. Error estimates are needed to determine when the accuracy is insufficient and a more costly special quadrature method should be utilized.
The final result of this paper are such quadrature error estimates for the composite Gauss-Legendre rule and the global trapezoidal rule, when applied to evaluate layer potentials defined over smooth curved surfaces in R^3. The estimates have no unknown coefficients and can be efficiently evaluated given the discretization of the surface, invoking a local one-dimensional root-finding procedure. They are derived starting with integrals over curves, using complex analysis involving contour integrals, residue calculus and branch cuts. By complexifying the parameter plane, the theory can be used to derive estimates also for curves in in R^3. These results are then used in the derivation of the estimates for integrals over surfaces. In this procedure, we also obtain error estimates for layer potentials evaluated over curves in R^2. Such estimates combined with a local root-finding procedure for their evaluation were earlier derived for the composite Gauss-Legendre rule for layer potentials written on complex form [4]. This is here extended to provide quadrature error estimates for both complex and real formulations of layer potentials, both for the Gauss-Legendre and the trapezoidal rule.
Numerical examples are given to illustrate the performance of the quadrature error estimates. The estimates for integration over curves are in many cases remarkably precise, and the estimates for curved surfaces in R^3 are also sufficiently precise, with sufficiently low computational cost, to be practically useful.
We study the effects of numerical quadrature rules on error convergence rates when solving Maxwell-type variational problems via the curl-conforming or edge finite element method. A complete {\em a priori} error analysis for the case of bounded polygonal and curved domains with non-homogeneous coefficients is provided. We detail sufficient conditions with respect to mesh refinement and precision for the quadrature rules so as to guarantee convergence rates following that of exact numerical integration. On curved domains, we isolate the error contribution to numerical quadrature rules.
Multilevel quadrature for elliptic parametric partial differential equations in case of polygonal approximations of curved domains
Universität Bonn（波恩大学）；Fraunhofer Institute for Algorithms and Scientific Computing (SCAI)（弗劳恩霍夫算法与科学计算研究所）；Universität Basel（巴塞尔大学）；Università della Svizzera italiana（瑞士意大利语大学）
This paper describes a trapezoidal quadrature method for the discretization of singular and hypersingular boundary integral operators (BIOs) that arise in solving boundary value problems for elliptic partial differential equations. The quadrature is based on a uniform grid in parameter space coupled with the standard punctured Trapezoidal rule. A key observation is that the error incurred by the singularity in the kernel can be expressed exactly using generalized Euler-Maclaurin formulae that involve the Riemann zeta function in 2D and the Epstein zeta functions in 3D. These expansions are exp…
The method of Helsing and co-workers evaluates Laplace and related layer potentials generated by a panel (composite) quadrature on a curve, efficiently and with high-order accuracy for arbitrarily close targets. Since it exploits complex analysis, its use has been restricted to two dimensions (2D). We first explain its loss of accuracy as panels become curved, using a classical complex approximation result of Walsh that can be interpreted as "electrostatic shielding" of a Schwarz singularity. We then introduce a variant that swaps the target singularity for one at its complexified parameter pr…
Numerical simulations with rigid particles, drops or vesicles constitute some examples that involve 3D objects with spherical topology. When the numerical method is based on boundary integral equations, the error in using a regular quadrature rule to approximate the layer potentials that appear in the formulation will increase rapidly as the evaluation point approaches the surface and the integrand becomes sharply peaked. To determine when the accuracy becomes insufficient, and a more costly special quadrature method should be used, error estimates are needed. In this paper we present quadratu…
We present algorithms for computing weakly singular and near-singular integrals arising when solving the 3D Helmholtz equation with curved boundary elements. These are based on the computation of the preimage of the singularity in the reference element's space using Newton's method, singularity subtraction, the continuation approach, and transplanted Gauss quadrature. We demonstrate the accuracy of our method for quadratic basis functions and quadratic triangles with several numerical experiments, including the scattering by two half-spheres.
In this paper, a Quadrature by Two Expansions (QB2X) numerical integration technique is developed for the single and double layer potentials of the Helmholtz equation in two dimensions. The QB2X method uses both local complex Taylor expansions and plane wave type expansions to achieve a resulting representation which is numerically accurate for all target points (interior, exterior, or exactly on the boundary) inside a leaf box in the fast multipole method (FMM) hierarchical tree structure. Compared to the original Quadrature by Expansion (QBX) method, the QB2X method explicitly includes the n…
Machine Learning Discovery of Optimal Quadrature Rules for Isogeometric Analysis
BCAM – Basque Center for Applied Mathematics（巴斯克应用数学中心（BCAM））；CUNEF University（CUNEF大学）；University of the Basque Country (UPV/EHU)（巴斯克大学（UPV/EHU））；Ikerbasque – Basque Foundation for Sciences（巴斯克科学基金会（Ikerbasque））
Pressure-robust finite elements for the Stokes problem on three-dimensional curved domains
LMAM and School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Chongqing Research Institute of Big Data, Peking University（北京大学重庆大数据研究院）；Institut für Mathematik, Friedrich-Schiller-Universität Jena（耶拿弗里德里希·席勒大学数学研究所）
This paper focuses on the approximation of continuous functions on the unit sphere by spherical polynomials of degree $n$ via hyperinterpolation. Hyperinterpolation of degree $n$ is a discrete approximation of the $L^2$-orthogonal projection of degree $n$ with its Fourier coefficients evaluated by a positive-weight quadrature rule that exactly integrates all spherical polynomials of degree at most $2n$. This paper aims to bypass this quadrature exactness assumption by replacing it with the Marcinkiewicz--Zygmund property proposed in a previous paper. Consequently, hyperinterpolation can be con…
Quadrature by Expansion (QBX) is a quadrature method for approximating the value of the singular integrals encountered in the evaluation of layer potentials. It exploits the smoothness of the layer potential by forming locally-valid expansion which are then evaluated to compute the near or on-surface value of the integral. Recent work towards coupling of a Fast Multipole Method (FMM) to QBX yielded a first step towards the rapid evaluation of such integrals (and the solution of related integral equations), albeit with only empirically understood error behavior. In this paper, we improve upon t…
Recursive reduction quadrature for the evaluation of Laplace layer potentials in three dimensions
Center for Computational Mathematics, Flatiron Institute, Simons Foundation（计算数学中心，Flatiron研究所，西蒙斯基金会）；Institute for Theoretical Sciences, Westlake University（理论科学研究所，西湖大学）
On Fast Implementation of Clenshaw-Curtis and Fejér-type Quadrature Rules
School of Mathematics and Statistics, Central South University（中南大学数学与统计学院）；School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）
Quadrature error estimates on non-matching grids in a fictitious domain framework for fluid-structure interaction problems
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Dipartimento di Matematica ‘F. Casorati’, Università degli Studi di Pavia（帕维亚大学 F. Casorati 数学系）；Istituto di Matematica Applicata e Tecnologie Informatiche ‘E. Magenes’, Consiglio Nazionale delle Ricerche（意大利国家研究委员会 E. Magenes 应用数学与信息技术研究所）；Dipartimento di Ingegneria Civile, Architettura, Territorio, Ambiente e di Matematica, Università degli Studi di Brescia（布雷西亚大学 土木、建筑、领土、环境与数学工程系）
Scaled boundary cubature scheme in higher dimensions: integration over polytopes and curved regions
Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；Department of Civil and Environmental Engineering, University of California, Davis（加州大学戴维斯分校土木与环境工程系）
This paper presents a novel method for solving partial differential equations on three-dimensional CAD geometries by means of immersed isogeometric discretizations that do not require quadrature schemes. It relies on a new developed technique for the evaluation of polynomial integrals over spline boundary representations that is exclusively based on analytical computations. First, through a consistent polynomial approximation step, the finite element operators of the Galerkin method are transformed into integrals involving only polynomial integrands. Then, by successive applications of the div…
Well-conditioned boundary integral methods for the solution of elliptic boundary value problems (BVPs) are powerful tools for static and dynamic physical simulations. When there are many close-to-touching boundaries (eg, in complex fluids) or when the solution is needed in the bulk, nearly-singular integrals must be evaluated at many targets. We show that precomputing a linear map from surface density to an effective source representation renders this task highly efficient, in the common case where each object is "simple", ie, its smooth boundary needs only moderately many nodes. We present a …
This article presents a new high-order accurate algorithm for finding a particular solution to a linear, constant-coefficient partial differential equation (PDE) by means of a convolution of the volumetric source function with the Green's function in complex geometries. Utilizing volumetric domain decomposition, the integral is computed over a union of regular boxes (lending the scheme compatibility with adaptive box codes) and triangular regions (which may be potentially curved near boundaries). Singular and near-singular quadrature is handled by converting integrals on volumetric regions to …
A three-grid high-order immersed finite element method for the analysis of CAD models
University of Glasgow（格拉斯哥大学）；Institute of Mathematics of the Czech Academy of Sciences（捷克科学院数学研究所）；Epic Games, Inc.（Epic Games公司）；University of Cambridge（剑桥大学）
When using boundary integral equation methods, we represent solutions of a linear partial differential equation as layer potentials. It is well-known that the approximation of layer potentials using quadrature rules suffer from poor resolution when evaluated closed to (but not on) the boundary. To address this challenge, we provide modified representations of the problem's solution. Similar to Gauss's law used to modify Laplace's double-layer potential, we use modified representations of Laplace's single-layer potential and Helmholtz layer potentials that avoid the close evaluation problem. So…
