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On a method to subtract off a singularity at a corner for the Dirichlet or Neumann problem

Author: Neil M. Wigley
Journal: Math. Comp. 23 (1969), 395-401
MSC: Primary 65.66
MathSciNet review: 0245223
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Abstract: Let $ D$ be a plane domain partly bounded by two line segments which meet at the origin and form there an interior angle $ \pi \alpha > 0$. Let $ U(x,y)$ be a solution in $ D$ of Poisson's equation such that either $ U$ or $ \partial U/\partial n$ (the normal derivative) takes prescribed values on the boundary segments. Let $ U(x,y)$ be sufficiently smooth away from the corner and bounded at the corner. Then for each positive integer $ N$ there exists a function $ {V_N}(x,y)$ which satisfies a related Poisson equation and which satisfies related boundary conditions such that $ U - {V_N}$ is $ N$-times continuously differentiable at the corner. If $ 1/\alpha $ is an integer $ {V_N}$ may be found explicitly in terms of the data of the problem for $ U$.

References [Enhancements On Off] (What's this?)

  • [1] Numerical solution of ordinary and partial differential equations., Based on a Summer School held in Oxford, August-September 1961, Pergamon Press, Oxford-London-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1962. MR 0146969
  • [2] S. Gerschgorin, ``Fehlerabschätzung für das Differenzenverfahren für Lösung partieller Differentialgleichungen,'' Z. Angew. Math. Mech., v. 10, 1930, pp. 373-382.
  • [3] Pentti Laasonen, On the behavior of the solution of the Dirichlet problem at analytic corners, Ann. Acad. Sci. Fenn. Ser. A. I. 1957 (1957), no. 241, 13. MR 0091405
  • [4] Hans Lewy, Developments at the confluence of analytic boundary conditions, Univ. California Publ. Math. (N.S.) 1 (1950), 247–280. MR 0040431
  • [5] R. Sherman Lehman, Developments at an analytic corner of solutions of elliptic partial differential equations, J. Math. Mech. 8 (1959), 727–760. MR 0105552
  • [6] W. R. Wasow, ``Asymptotic development of the solution of Dirichlet's problem at analytic corners,'' Duke Math. J., v. 24, 1957, pp. 47-56. MR 18, 568.
  • [7] Neil M. Wigley, Asymptotic expansions at a corner of solutions of mixed boundary value problems, J. Math. Mech. 13 (1964), 549–576. MR 0165227
  • [8] George E. Forsythe and Wolfgang R. Wasow, Finite-difference methods for partial differential equations, Applied Mathematics Series, John Wiley & Sons, Inc., New York-London, 1960. MR 0130124

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Article copyright: © Copyright 1969 American Mathematical Society

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