A highorder integral algorithm for highly singular PDE solutions in Lipschitz domains
 Oscar P. Bruno,
 Jeffrey S. Ovall,
 Catalin Turc
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We present a new algorithm, based on integral equation formulations, for the solution of constantcoefficient elliptic partial differential equations (PDE) in closed twodimensional domains with nonsmooth boundaries; we focus on cases in which the integralequation solutions as well as physically meaningful quantities (such as, stresses, electric/magnetic fields, etc.) tend to infinity at singular boundary points (corners). While, for simplicity, we restrict our discussion to integral equations associated with the Neumann problem for the Laplace equation, the proposed methodology applies to integral equations arising from other types of PDEs, including the Helmholtz, Maxwell, and linear elasticity equations. Our numerical results demonstrate excellent convergence as discretizations are refined, even around singular points at which solutions tend to infinity. We demonstrate the efficacy of this algorithm through applications to solution of Neumann problems for the Laplace operator over a variety of domains—including domains containing extremely sharp concave and convex corners, with angles as small as π/100 and as large as 199π/100.
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 Title
 A highorder integral algorithm for highly singular PDE solutions in Lipschitz domains
 Journal

Computing
Volume 84, Issue 34 , pp 149181
 Cover Date
 20090601
 DOI
 10.1007/s0060700900311
 Print ISSN
 0010485X
 Online ISSN
 14365057
 Publisher
 Springer Vienna
 Additional Links
 Topics
 Keywords

 Boundary value problems
 Secondkind integral equations
 Singular solution
 Highorder methods
 31A05
 35C15
 65N35
 65R20
 Industry Sectors
 Authors

 Oscar P. Bruno ^{(1)}
 Jeffrey S. Ovall ^{(1)}
 Catalin Turc ^{(2)}
 Author Affiliations

 1. California Institute of Technology, Pasadena, CA, 91125, USA
 2. Case Western Reserve University, Cleveland, OH, 44106, USA