Summary
For solving Laplace's boundary value problems with singularities, a nonconforming combined approach of the Ritz-Galerkin method and the finite element method is presented. In this approach, singular functions are chosen to be admissible functions in the part of a solution domain where there exist singularities; and piecewise linear functions are chosen to be admissible functions in the rest of the solution domain. In addition, the admissible functions used here are constrained to be continuous only at the element nodes on the common boundary of both methods. This method is nonconforming; however, the nonconforming effect does not result in larger errors of numerical solutions as long as a suitable coupling strategy is used.
In this paper, we will develop such an approach by using a new coupling strategy, which is described as follows: IfL+1=O(|lnh|), the average errors of numerical solutions and their generalized derivatives are stillO(h), whereh is the maximal boundary length of quasiuniform triangular elements in the finite element method, andL+1 is the total number of singular admissible functions in the Ritz-Galerkin method. The coupling relation,L+1=O(|lnh|), is significant because only a few singular functions are required for a good approximation of solutions.
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References
Babuška, I., Aziz, A.K.: Survey lectures on the mathematical foundations of the finite element method. In: The Mathematical Foundations of the Finite Element Method with Application to Partial Differential Equations (A.K. Aziz, ed.), pp. 3–359. New York, London: Academic Press 1972
Brebbia, C.A.: The Boundary Element Method for Engineering. London: Pentch Press 1978
Ciarlet, P.G.: The Finite Element Method for Elliptic Problem. Amsterdam, New York, Oxford: North Holland 1978
Courant, R., Hilbert, D.: Methods of Mathematical Physics. Vol. 1. New York: Interscience 1953
Delves, L.M., Hall, C.A.: An implicit matching principle for global element calculation. J. Inst. Math. Appl.23, 223–234 (1979)
Delves, L.M.: Global and regional methods. In: A Survey of Numerical Methods for Partial Differential Equations (Gladwell, L. and Wait, R. eds.). pp. 106–127. Oxford: Clarendon 1979
Fox, L.: Finite differences and singularities on elliptic problem. In: A Survey of Numerical Methods for Partial Differential Equations. (Gladwell, L. and Wait, R. eds.), pp. 42–69. Oxford: Clarendon 1979
Kishimoto, K., Yamaguchi, I., Tachihara, M., Aoki, S., Sakata, M.: Elastic-plastic fracture mechanics analysis by combination of boundary and finite element method. In: Boundary Element (Brebbie, C.A., Futagami, T., and Tanaka, M. eds.), pp. 975–984. Berlin, Heildelberg New York, Tokyo: Springer 1983
Lehman, R.S.: Developments near an analytic corner of solution of elliptic differential equation. J. Math. Mech.8, 727–760 (1959)
Li, Z.C.: The combined method between original energy method and finite element method for solving Laplace's boundary value with singularity. (in Chinese). Math. Numer. Sin.2, 319–328 (1980)
Li, Z.C.: An approach for combining the Ritz-Galerkin and finite element methods. J. Approximation Theory39, 132–152 (1983)
Li, Z.C., Liang, G.P.: On the combined methods of the boundary value problems of elliptic equations. (in Chinese). Math. Numer. Sin.2, 192–194 (1980)
Li, Z.C., Liang, G.P.: On Ritz-Galerkin-F.E.M. combined method of solving the boundary value problems of elliptic equations. Sci. Sin.24, 1497–1508 (1981)
Li, Z.C., Liang, G.P.: On the simplified hybrid-combined method. Math. Comput.41, 13–25 (1983)
Li, Z.C., Mathon, R., Sermer, P.: Boundary methods for solving elliptic problems with singularities and interfaces. (To appear in SIAM J. Numer. Anal.)
Liang, G.P., Gu, Y.G.: The semi-analytic method for the plane plastic problems with cracks. (in Chinese). Math. Numer. Sin.4, 65–78 (1978)
Han, H.: The numerical solution of interface problems in infinite element methods. Numer. Math.39, 39–50 (1982)
Schartz, A.H., Wablbin, L.B.: Maximum norm estimaties in the finite element method on plan polygonal domain. Part I. Math. Comput.32, 73–109 (1978)
Scott, R.: OptimalL ∞ estimates for the finite element method on irregular meshes. Math. Comput.30, 681–697 (1976)
Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. New Jersey: Prentice-Hall, Englewood Cliffs 1973
Thatcher, R.W.: The use of infinite grid refinement at singularities in the solution of Laplace's equation. Numer. Math.25, 163–178 (1976)
Whiteman, J.R., Papmichael, N.: Treatment of harmonic mixed boundary problems by conformal transformation methods. Z.A.M.P.23, 655–664 (1972)
Wigley, N.M.: Asymptotic expansions at a corner of solutions of mixed boundary value problems. J. Math. Mech.4, 549–576 (1964)
Wigley, N.M.: On a method to subtract off a singularity at corner for the Dirichlet or Neuman problem. Math. Comput.23, 395–401 (1969)
Ying, L.: The infinite similar element method for calculating stress intensity factors. Sci. Sin.21, 19–43 (1978)
Zienkiewciz, C., Kelley, D.W., Bettess, P.: The coupling of the finite element method and boundary solution procedures. Intern. J. Numer. Methods Engrg.11, 355–375 (1977)
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This material is from Chapter 5 in my Ph.D. thesis: Numerical Methods for Elliptic Boundary Value Problems with Singularities. Part I: Boundary Methods for Solving Elliptic Problems with Singularities. Part II: Nonconforming Combinations for Solving Elliptic Problems with Singularities, the Department of Mathematics and Applied Mathematics, University of Toronto, May 1986
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Li, ZC. A nonconforming combined method for solving Laplace's boundary value problems with singularities. Numer. Math. 49, 475–497 (1986). https://doi.org/10.1007/BF01389701
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DOI: https://doi.org/10.1007/BF01389701