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28
Corner singularities for elliptic problems: integral equations, graded meshes, quadrature, and . . .
, 2009
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Integral equation methods for elliptic problems with boundary conditions of mixed type
 J. Comput. Phys
"... of mixed typeI ..."
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Quadrature by Expansion: A New Method for the Evaluation of Layer Potentials
, 2012
"... Integral equation methods for the solution of partial differential equations, when coupled with suitable fast algorithms, yield geometrically flexible, asymptotically optimal and wellconditioned schemes in either interior or exterior domains. The practical application of these methods, however, req ..."
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Cited by 23 (8 self)
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Integral equation methods for the solution of partial differential equations, when coupled with suitable fast algorithms, yield geometrically flexible, asymptotically optimal and wellconditioned schemes in either interior or exterior domains. The practical application of these methods, however, requires the accurate evaluation of boundary integrals with singular, weakly singular or nearly singular kernels. Historically, these issues have been handled either by loworder product integration rules (computed semianalytically), by singularity subtraction/cancellation, by kernel regularization and asymptotic analysis, or by the construction of special purpose “generalized Gaussian quadrature ” rules. In this paper, we present a systematic, highorder approach that works for any singularity (including hypersingular kernels), based only on the assumption that the field induced by the integral operator is locally smooth when restricted to either the interior or the exterior. Discontinuities in the field across the boundary are permitted. The scheme, denoted QBX (quadrature by expansion), is easy to implement and compatible with fast hierarchical algorithms such as the fast multipole method. We include accuracy tests for a variety of integral operators in two dimensions on smooth and corner domains.
A fast and stable solver for singular integral equations on piecewise smooth curves
 SIAM J. Sci. Comput
"... Abstract. A scheme for the numerical solution of singular integral equations on piecewise smooth curves is presented. It relies on several techniques: reduction, Nyström discretization, composite quadrature, recursive compressed inverse preconditioning, and multipole acceleration. The scheme is fa ..."
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Cited by 19 (11 self)
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Abstract. A scheme for the numerical solution of singular integral equations on piecewise smooth curves is presented. It relies on several techniques: reduction, Nyström discretization, composite quadrature, recursive compressed inverse preconditioning, and multipole acceleration. The scheme is fast and stable. Its computational cost grows roughly logarithmically with the precision sought and linearly with overall system size. When the integral equation models a boundary value problem, the achievable accuracy may be close to the condition number of that problem times machine epsilon. This is illustrated by application to elastostatic problems involving zigzagshaped cracks with up to twenty thousand corners and branched cracks with hundreds of triple junctions.
THE EXPONENTIALLY CONVERGENT TRAPEZOIDAL RULE
"... Abstract. It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods ..."
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Cited by 17 (3 self)
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Abstract. It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.
Solving integral equations on piecewise smooth boundaries using the RCIP method: a tutorial
, 2015
"... Recursively Compressed Inverse Preconditioning (RCIP) is a numerical method for obtaining highly accurate solutions to integral equations on piecewise smooth surfaces. The method originated in 2008 as a technique within a scheme for solving Laplace’s equation in twodimensional domains with corners. ..."
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Cited by 14 (9 self)
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Recursively Compressed Inverse Preconditioning (RCIP) is a numerical method for obtaining highly accurate solutions to integral equations on piecewise smooth surfaces. The method originated in 2008 as a technique within a scheme for solving Laplace’s equation in twodimensional domains with corners. In a series of subsequent papers the technique was then refined and extended as to apply to integral equation formulations of a broad range of boundary value problems in physics and engineering. The purpose of the present paper is threefold: First, to review the RCIP method in a simple setting. Second, to show how easily the method can be implemented in Matlab. Third, to present new applications of RCIP to integral equations of scattering theory on planar curves with corners.
EVALUATION OF LAYER POTENTIALS CLOSE TO THE BOUNDARY FOR LAPLACE AND HELMHOLTZ PROBLEMS ON ANALYTIC PLANAR DOMAINS
"... Abstract. Boundary integral equations are an efficient and accurate tool for the numerical solution of elliptic boundary value problems. The solution is expressed as a layer potential; however, the error in its evaluation grows large near the boundary if a fixed quadrature rule is used. Firstly, we ..."
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Cited by 9 (2 self)
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Abstract. Boundary integral equations are an efficient and accurate tool for the numerical solution of elliptic boundary value problems. The solution is expressed as a layer potential; however, the error in its evaluation grows large near the boundary if a fixed quadrature rule is used. Firstly, we analyze this error for Laplace’s equation with analytic density and the global periodic trapezoid rule, and find an intimate connection to the complexification of the boundary parametrization. Our main result is a simple and efficient scheme for accurate evaluation up to the boundary for single and doublelayer potentials for the Laplace and Helmholtz equations, using surrogate local expansions about centers placed near the boundary. The scheme—which also underlies the recent QBX Nyström quadratures—is exponentially convergent (we prove this in the analytic Laplace case), requires no adaptivity, and has O(N) complexity when executed via a locallycorrected fast multipole sum. We give an example of highfrequency scattering from an obstacle with perimeter 700 wavelengths, evaluating the solution at 2 × 10 5 points near the boundary with 11digit accuracy in 30 seconds in MATLAB on a single CPU core.
Elastostatic computations on aggregates of grains with sharp interfaces, corners, and triplejunctions
 Journal of Solids Structures
"... and triplejunctions ..."
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An explicit kernelsplit panelbased Nyström scheme for integral equations on axially symmetric surfaces
 J. Comput. Phys
, 2014
"... boundary value problems ..."