Abstract
The key property of elliptic systems is that their solution tends to be as smooth as the data and other factors permit. This condition strikingly constrasts with hyperbolic systems, in which singular behavior (e.g., shocks) can arise even if all inputs are smooth.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
J.L. Swedlow (ed.), The Surface Crack, ASME, New York, November, 1972.
G.R. Irwin, Fracture Mechanics, Pergamon, Elmsford, NY, 1960.
N.I. Muskhelishvili, Some Problems of the Mathematical Theory of Elasticity, North-Holland, Groningen, 1953.
A.E.H. Love, The Mathematical Theory of Elasticity, Cambridge University Press, Cambridge, UK, 1927.
A.A. Griffith, “The phenomenon of rapture and flow in solids,” Philos. Trans. Roy. Soc. Ser. (A), 221(1920), 163 – 179.
G.R. Irwin, “Analysis of stresses and strains near the end of a crack,” J. Appl. Mech., 24(1957).
Milne-Thomson, Theoretical Hydrodynamics, Macmillan, New York, 1968.
G. Birkhoff, Numerical Solution of Elliptic Equations, SIAM, Philadelphia, 1971.
I. Babuska and B. Kellogg, The Mathematical Foundation of the Finite Element Method, (A.K. Aziz, ed.), Academic Press, New York, 1972.
Z. Schuss, Theory and Applications of Stochastic Differential Equations, Wiley, New York, 1980.
J.L. Zeman, Approximate Analysis of Schochastic Processes in Mechanics, Springer- Verlag, New York, 1971.
G. Strang and G.J. Fix, An Analysis of the Finite Element Method, Academic Press, New York, 1973.
J.F. Thompson, ZUA Warsi and C.W. Mastin, Numerical Grid Generation, North-Holland, Amsterdam, 1985.
K. Miller and N. Miller, “Moving finite elements,” SIAM J. Numer. Anal, 18(1981), 1019.
I. Babuska and W. Rheinboldt, “Error estimates for adaptive finite element computations,” SIAM J. Numer. Anal, 15(1978).
I. Babuska and G. Gus, “The h-pversion of the finite element method,” Tech. Note BN-1043, Institute for Physical Science and Technology, University of Maryland, College Park, 1985.
V.A. Kondrat’ev, “Boundary value problems for elliptic equations in domains with conic or angular points,” Trans. Moscow Math. Soc., 69(1967), 227 – 313.
V.A. Kondrat’ev and O.A. Oleinik, “Boundary value problems for partial differential equations in nonsmooth domains,” Russian Math. Surveys, 38, No. 2, (1983), 1 – 86.
L. Williams, “Stress singularities resulting from various boundary conditions in angular corners,” J. Appl Mech., 19(1952), 526 – 528.
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.
P. Tolksdorf, “On the behavior near the boundary of solutions of quasilinear equations,” Preprint No. 459, Universität Bonn, 1981.
L. Lehman, “Developments at an analytic corner of solutions of elliptic partial differential equations,” J. Math. Mech., 8 (1959), 727 – 760.
I. Babuška, R.B. Kellogg, and Pitkaranta, “Direct and inverse error estimates for finite elements with mesh refinements,” Numer. Math., 33(1979), 447 – 471.
C.L. Cox and G.J. Fix, “On the accuracy of least square methods in the presence of corner singularities,” Comput. Math. Appl, 10(1984), 463 – 475.
G.J. Fix, S. Gulati, and G.I. Wakoff, “On the use of singular functions with finite element approximations,” J. Comput. Phys., 13(1973), 209 – 228.
Y. Lee, “Shear bands in elastic-perfectly plastic materials,” Ph.D. thesis, Carnegie Mellon University, 1981.
G.I. Wakoff, “Piecewise polynomial spaces and the Ritz—Galerkin method,” Ph.D. thesis, Harvard University, 1970.
R. Wait and A.R. Mitchell, “Corner singularities in elliptic problems,” J. Comput. Phys., 8 (1971), 45– 52.
E. Byskov, “Calculation of stress intensity factors using finite element methods,” Internat. J. Fracture, 6(1976), 159 – 168.
W.S. Blackburn, “Calculation of stress intensity factors at crack tips using special finite elements,” in Mathematics of Finite Elements and Applications( J.R. Whiteman, ed.), Academic Press, London, 1973.
H. Blum and M. Dobrowolski, “On finite element methods for elliptic equations on domains with corners,” Computing, 25(1983), 53 – 63.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1988 Springer Science+Business Media New York
About this paper
Cite this paper
Fix, G.J. (1988). Singular Finite Element Methods. In: Dwoyer, D.L., Hussaini, M.Y., Voigt, R.G. (eds) Finite Elements. ICASE/NASA LaRC Series. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-3786-0_3
Download citation
DOI: https://doi.org/10.1007/978-1-4612-3786-0_3
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4612-8350-8
Online ISBN: 978-1-4612-3786-0
eBook Packages: Springer Book Archive