# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A high-order integral algorithm for highly singular PDE solutions in Lipschitz domains. (English) Zbl 1176.65139
Summary: We present a new algorithm, based on integral equation formulations, for the solution of constant-coefficient elliptic partial differential equations (PDE) in closed two-dimensional domains with non-smooth boundaries; we focus on cases in which the integral-equation solutions as well as physically meaningful quantities (such as, stresses, electric/magnetic fields, etc.) tend to infinity at the 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 $\pi /100$ and as large as $199\pi /100$.

##### MSC:
 65N38 Boundary element methods (BVP of PDE) 35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation 35C15 Integral representations of solutions of PDE 65N12 Stability and convergence of numerical methods (BVP of PDE)
Full Text:
##### References:
 [1] Atkinson KE (1997) The numerical solution of integral equations of the second kind. Cambridge monographs on applied and computational mathematics , vol 4. Cambridge University Press, Cambridge · Zbl 0899.65077 [2] Atkinson KE, Graham IG (1988) An iterative variant of the Nyström method for boundary integral equations on nonsmooth boundaries. The mathematics of finite elements and applications, VI (Uxbridge, 1987), London, pp 297--303 [3] Babuška I, Kellogg RB, Pitkäranta J (1979) Direct and inverse error estimates for finite elements with mesh refinements. Numer Math 33: 447--471 · Zbl 0423.65057 · doi:10.1007/BF01399326 [4] Babuška I, von Petersdorff T, Andersson B (1994) Numerical treatment of vertex singularities and intensity factors for mixed boundary value problems for the Laplace equation in R 3. SIAM J Numer Anal 31: 1265--1288 · Zbl 0806.65107 · doi:10.1137/0731066 [5] Babuška I, Miller A (1984) The post-processing approach in the finite-element method, 2: The calculation of stress intensity factors. Int J Numer Methods Eng 20: 1111--1129 · Zbl 0535.73053 · doi:10.1002/nme.1620200611 [6] Björck Å (1996) Numerical methods for least squares problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia · Zbl 0858.65043 [7] Bleszynski E, Bleszynski M, Jaroszewicz T (1996) AIM: adaptive integral method for solving large-scale electromagnetic scattering and radiation problems. Radio Sci 31: 1225--1251 · Zbl 1071.78015 · doi:10.1029/96RS02504 [8] Borsuk M, Kondrat’ev V (2006) Elliptic boundary value problems of second order in piecewise smooth domains, vol 69. North-Holland Mathematical Library. Elsevier, Amsterdam (2006) [9] Bruno OP, Kunyansky LA (2001) A fast, high-order algorithm for the solution of surface scattering problems: basic implementation, tests, and applications. J Comput Phys 169: 80--110 · Zbl 1052.76052 · doi:10.1006/jcph.2001.6714 [10] Costabel M, Ervin VJ, Stephan EP (1993) Quadrature and collocation methods for the double layer potential on polygons. Z Anal Anwendungen 12: 699--707 · Zbl 0791.65087 [11] Costabel M, Stephan E (1983) Curvature terms in the asymptotic expansions for solutions of boundary integral equations on curved polygons. J Integr Equ 5: 353--371 · Zbl 0538.35022 [12] Costabel M, Stephan E (1983) The normal derivative of the double layer potential on polygons and Galerkin approximation. Appl Anal 16: 205--228 · Zbl 0515.35036 · doi:10.1080/00036818308839470 [13] Costabel M, Stephan E (1985) Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation. Mathematical models and methods in mechanics, Banach Center Publ. 15, Warsaw, pp 175--251 · Zbl 0655.65129 [14] Costabel M, Dauge M (2000) Singularities of electromagnetic fields in polyhedral domains. Arch Ration Mech Anal 151: 221--276 · Zbl 0968.35113 · doi:10.1007/s002050050197 [15] Cox C, Fix G (1985) On the accuracy of least squares methods in the presence of corner singularities. Comput Math Appl 10: 463--475 · Zbl 0573.65081 · doi:10.1016/0898-1221(84)90077-4 [16] Demkowicz L, Devloo P, Oden J (1985) On an h-type mesh-refinement strategy based on minimization of interpolation errors. Comput Methods Appl Mech Eng 53: 67--89 · Zbl 0556.73081 · doi:10.1016/0045-7825(85)90076-3 [17] Devloo P (1990) Recursive elements, an inexpensive solution process for resolving point singularities in elliptic problems. In: Proceedings of 2nd World Congress on Computational Mechanics. IACM, Stuttgart, pp 609--612 [18] Elschner J, Jeon Y, Sloan IH, Stephan EP (1997) The collocation method for mixed boundary value problems on domains with curved polygonal boundaries. Numer Math 76: 355--381 · Zbl 0878.65102 · doi:10.1007/s002110050267 [19] Elschner J, Stephan EP (1996) A discrete collocation method for Symm’s integral equation on curves with corners. J Comput Appl Math 75: 131--146 · Zbl 0868.65098 · doi:10.1016/S0377-0427(96)00070-2 [20] Fix GJ, Gulati S, Wakoff GI (1973) On the use of singular functions with finite element approximations. J Comput Phys 13: 209--228 · Zbl 0273.35004 · doi:10.1016/0021-9991(73)90023-5 [21] Ganesh M, Graham IG (2004) A high-order algorithm for obstacle scattering in three dimensions. J Comput Phys 198: 211--242 · Zbl 1052.65108 · doi:10.1016/j.jcp.2004.01.007 [22] Graham IG, Chandler GA (1988) High-order methods for linear functionals of solutions of second kind integral equations. SIAM J Numer Anal 25: 1118--1137 · Zbl 0661.65137 · doi:10.1137/0725064 [23] Grisvard P (1985) Elliptic problems in nonsmooth domains. Monographs and Studies in Mathematics, vol 24. Pitman (Advanced Publishing Program), Boston · Zbl 0695.35060 [24] Grisvard P (1992) Singularities in boundary value problems. Recherches en Mathématiques Appliquées (Research in Applied Mathematics), vol 22. Masson, Paris · Zbl 0766.35001 [25] Givoli D, Rivkin L, Keller JB (1992) A finite element method for domains with corners. Int J Numer Methods Eng 35: 1329--1345 · Zbl 0768.73072 · doi:10.1002/nme.1620350611 [26] Heuer N, Mellado ME, Stephan EP (2002) A p-adaptive algorithm for the BEM with the hypersingular operator on the plane screen. Int J Numer Methods Eng 53: 85--104 · Zbl 0996.65128 · doi:10.1002/nme.393 [27] Heuer N, Stephan EP (1996) The hp-version of the boundary element method on polygons. J Int Equations Appl 8: 173--212 · Zbl 0888.65121 · doi:10.1216/jiea/1181075935 [28] Heuer N, Stephan EP (1998) Boundary integral operators in countably normed spaces. Math Nachr 191: 123--151 · Zbl 0898.65078 · doi:10.1002/mana.19981910107 [29] Hughes T, Akin J (1980) Techniques for developing ’special’ finite element shape function with particular reference to singularities. Int J Numer Methods Eng 15: 733--751 · Zbl 0428.73074 · doi:10.1002/nme.1620150509 [30] Hunter DB, Smith HV (2005) A quadrature formula of Clenshaw--Curtis type for the Gegenbauer weight-function. J Comput Appl Math 177: 389--400 · Zbl 1090.41011 · doi:10.1016/j.cam.2004.08.014 [31] Jerison DS, Kenig CE (1981) The Neumann problem on Lipschitz domains. Bull Am Math Soc (N.S.) 4: 203--207 · Zbl 0471.35026 · doi:10.1090/S0273-0979-1981-14884-9 [32] John F (1991) Partial differential equations, vol 1. Springer, New York [33] Kondrat’ VA (1967) Boundary value problems for elliptic equations in domains with conical or angular points. Trans Moscow Math Soc 16: 227--313 [34] Kozlov VA, Maz’ VG, Rossmann J (1997) Elliptic boundary value problems in domains with point singularities. Mathematical surveys and monographs, vol 52. American Mathematical Society, Providence [35] Kress R (1989) Linear integral equations, vol 82. Springer, Berlin · Zbl 0671.45001 [36] Kress R (1990) A Nyström method for boundary integral equations in domains with corners. Numer Math 58: 145--161 · Zbl 0707.65078 · doi:10.1007/BF01385616 [37] Lin K, Tong P (1980) Singular finite elements for the fracture analysis of V-notched plate. Int J Numer Methods Eng 15: 1343--1354 · Zbl 0441.73098 · doi:10.1002/nme.1620150907 [38] Maischak M, Stephan EP (1997) The hp-version of the boundary element method in R 3: the basic approximation results. Math Methods Appl Sci 20: 461--476 · Zbl 0872.65101 · doi:10.1002/(SICI)1099-1476(19970325)20:5<461::AID-MMA819>3.0.CO;2-X [39] Maue A-W (1949) Zur Formulierung eines allgemeinen Beugungsproblems durch eine Integralgleichung. Z Phys 126: 601--618 · Zbl 0033.14101 · doi:10.1007/BF01328780 [40] Monegato G, Scuderi L (2003) A polynomial collocation method for the numerical solution of weakly singular and nonsingular integral equations on non-smooth boundaries. Int J Numer Methods Eng 58: 1985--2011 · Zbl 1035.65145 · doi:10.1002/nme.843 [41] Rokhlin V (1993) Diagonal forms of translation operators for the Helmholtz equation in three dimensions. Appl Comput Harmon Anal 1: 82--93 · Zbl 0795.35021 · doi:10.1006/acha.1993.1006 [42] Sloan IH, Smith WE (1980) Product integration with the Clenshaw--Curtis points: implementation and error estimates. Numer Math 34: 387--401 · Zbl 0416.65014 · doi:10.1007/BF01403676 [43] Song J, Lu C, Chew W, Lee S (1998) Fast Illinois Solver Code (FISC). IEEE Antennas Propag Mag 40: 27--34 · doi:10.1109/74.706067 [44] Stern M (1979) Families of consistent conforming elements with singular derivative fields. Int J Numer Methods Eng 14: 409--421 · Zbl 0408.73071 · doi:10.1002/nme.1620140307 [45] Strain J (1995) Locally corrected multidimensional quadrature rules for singular functions. SIAM J Sci Comput 16: 992--1017 · Zbl 0828.65016 · doi:10.1137/0916058 [46] Trefethen LN (2008) Is Gauss Quadrature Better than Clenshaw-Curtis?. SIAM Rev 50: 67--87 · Zbl 1141.65018 · doi:10.1137/060659831 [47] Weideman JAC, Trefethen LN (2007) The kink phenomenon in Fejér and Clenshaw-Curtis quadrature. Numer Math 107: 707--727 · Zbl 1142.41010 · doi:10.1007/s00211-007-0101-2 [48] Verchota G (1984) Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J Funct Anal 59: 572--611 · Zbl 0589.31005 · doi:10.1016/0022-1236(84)90066-1 [49] Wigley NM (1964) Asymptotic expansions at a corner of solutions of mixed boundary value problems. J Math Mech 13: 549--576 · Zbl 0178.45902 [50] Wigley NM (1969) On a method to subtract off a singularity at a corner for the Dirichlet or Neumann problem. Math Comput 23: 395--401 · Zbl 0176.15802 · doi:10.1090/S0025-5718-1969-0245223-0 [51] Wigley NM (1987) Stress intensity factors and improved convergence estimates at a corner. SIAM J Numer Anal 24: 350--354 · Zbl 0622.65112 · doi:10.1137/0724026 [52] Wu X, Xue W (2003) Discrete boundary conditions for elasticity problems with singularities. Comput Methods Appl Mech Eng 192: 3777--3795 · Zbl 1054.74007 · doi:10.1016/S0045-7825(03)00375-X [53] Wu X, Han H (1997) A finite-element method for Laplace- and Helmholtz-type boundary value problems with singularities. SIAM J Numer Anal 34: 1037--1050 · Zbl 0873.65100 · doi:10.1137/S0036142993258488 [54] Zargaryan SS, Maz’ VG (1984) The asymptotic form of the solutions of integral equations of potential theory in the neighbourhood of the corner points of a contour. Prikl Mat Mekh 48: 169--174