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

DOI:
https://doi.org/10.1090/S0025-5718-1969-0245223-0

MathSciNet review:
0245223

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Abstract: Let be a plane domain partly bounded by two line segments which meet at the origin and form there an interior angle . Let be a solution in of Poisson's equation such that either or (the normal derivative) takes prescribed values on the boundary segments. Let be sufficiently smooth away from the corner and bounded at the corner. Then for each positive integer there exists a function which satisfies a related Poisson equation and which satisfies related boundary conditions such that is -times continuously differentiable at the corner. If is an integer may be found explicitly in terms of the data of the problem for .

**[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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DOI:
https://doi.org/10.1090/S0025-5718-1969-0245223-0

Article copyright:
© Copyright 1969
American Mathematical Society