We present a simple nonconforming approximation of the linear two–point boundary value problem which violates patch test requirements. Nevertheless the solutions, obtained from these type of approximations, converge to the exact solution.
[1] G.P. Bazeley, Y.K. Cheung, B.M. Irons, O.C. Zienkiewicz, Triangular
elements in bending: Conforming and nonconforming solutions in Proceedings
Conference on Matrix Methods in Structural Mechanics, Wright
Patterson A. F. B., Dayton, OH, 547-576.
[2] B.M. Irons, A. Razzaque, Experience with the patch test for convergence
of finite elements in A. K. Aziz, ed., The mathematical Foundations
of the Finite Element Method with Applications to Partial Differential
Equations, Academic Press, New York, 557-587.
[3] G. Strang, G.J. Fix, An analysis of the finite element method, Prentice-
Hall Inc., Englewood Cliffs, N.J., 1973.
[4] G. Sander, P. Beckers The influence of the choice of connectors in the
finite element method, Proc. conf. on the mathematical aspects of the finite
element methods, Rome 75, Galligani, I., Magenes, E., eds., Lecture notes
in mathematics, 606, Springer-Verlag, New York, 316-340, 1977.
[5] E.R.A. Oliveira, The patch test and the general convergence criteria of
the finite element method, International journal of solids and structures,
vol. 13, 159-178, 1977.
[6] F. Stummel, The generalised patch test, SIAM Journal of numerical
analysis, vol. 16, no. 3, 449-471, 1979.
[7] F. Stummel, The limitations of the patch test, International journal for
numerical methods in engineering, vol. 15, 177-188, 1980.
[8] B.M. Irons, M. Loikkanen, An engineer-s defence of the patch test,
International journal for numerical methods in engineering, vol. 19, 1391-
1401, 1983.
[9] Z.C. Shi, An explicit analysis of Stummel-s patch test examples, International
journal for numerical methods in engineering, vol. 20, 1233-1246,
1984.
[10] Z.C. Shi, A convergence condition for the quadrilateral Wilson element,
Numerische mathematik, vol. 44, 394-361, 1984.
[11] Z.C. Shi, The F-E-M-test for convergence of nonconforming finite
elements, Mathematics of computation, vol. 49, no. 40, 391-405, 1987.
[12] Z.C. Shi, On Stummel-s examples to the patch test, Computational
mechanics, vol. 5, 81-87 (1989).
[13] R.L. Taylor, J.C. Simo, O.C. Zienkiewicz, A.C.H. Chan, The patch test-
a condition for assessing FEM convergence, International journal for
numerical methods in engineering, vol. 22, 39-62, 1986.
[14] O.C. Zienkiewicz, R.L. Taylor, The finite element patch test revisited. A
computer test for convergence, validation and error estimates, Computer
methods in applied mechanics and engineering, 149, 223-254, 1997.
[15] P.G. Ciarlet, J.L. Lions, Handbook of numerical analysis, Volume II,
Finite element methods (Part 1), North-Holland, Amsterdam, 1991.
[16] M. Wang, On the necessity and sufficiency of the patch test for
convergence of nonconforming finite elements, SIAM J. Numer. Anal.,
39(2), 363-384, 2001.
[1] G.P. Bazeley, Y.K. Cheung, B.M. Irons, O.C. Zienkiewicz, Triangular
elements in bending: Conforming and nonconforming solutions in Proceedings
Conference on Matrix Methods in Structural Mechanics, Wright
Patterson A. F. B., Dayton, OH, 547-576.
[2] B.M. Irons, A. Razzaque, Experience with the patch test for convergence
of finite elements in A. K. Aziz, ed., The mathematical Foundations
of the Finite Element Method with Applications to Partial Differential
Equations, Academic Press, New York, 557-587.
[3] G. Strang, G.J. Fix, An analysis of the finite element method, Prentice-
Hall Inc., Englewood Cliffs, N.J., 1973.
[4] G. Sander, P. Beckers The influence of the choice of connectors in the
finite element method, Proc. conf. on the mathematical aspects of the finite
element methods, Rome 75, Galligani, I., Magenes, E., eds., Lecture notes
in mathematics, 606, Springer-Verlag, New York, 316-340, 1977.
[5] E.R.A. Oliveira, The patch test and the general convergence criteria of
the finite element method, International journal of solids and structures,
vol. 13, 159-178, 1977.
[6] F. Stummel, The generalised patch test, SIAM Journal of numerical
analysis, vol. 16, no. 3, 449-471, 1979.
[7] F. Stummel, The limitations of the patch test, International journal for
numerical methods in engineering, vol. 15, 177-188, 1980.
[8] B.M. Irons, M. Loikkanen, An engineer-s defence of the patch test,
International journal for numerical methods in engineering, vol. 19, 1391-
1401, 1983.
[9] Z.C. Shi, An explicit analysis of Stummel-s patch test examples, International
journal for numerical methods in engineering, vol. 20, 1233-1246,
1984.
[10] Z.C. Shi, A convergence condition for the quadrilateral Wilson element,
Numerische mathematik, vol. 44, 394-361, 1984.
[11] Z.C. Shi, The F-E-M-test for convergence of nonconforming finite
elements, Mathematics of computation, vol. 49, no. 40, 391-405, 1987.
[12] Z.C. Shi, On Stummel-s examples to the patch test, Computational
mechanics, vol. 5, 81-87 (1989).
[13] R.L. Taylor, J.C. Simo, O.C. Zienkiewicz, A.C.H. Chan, The patch test-
a condition for assessing FEM convergence, International journal for
numerical methods in engineering, vol. 22, 39-62, 1986.
[14] O.C. Zienkiewicz, R.L. Taylor, The finite element patch test revisited. A
computer test for convergence, validation and error estimates, Computer
methods in applied mechanics and engineering, 149, 223-254, 1997.
[15] P.G. Ciarlet, J.L. Lions, Handbook of numerical analysis, Volume II,
Finite element methods (Part 1), North-Holland, Amsterdam, 1991.
[16] M. Wang, On the necessity and sufficiency of the patch test for
convergence of nonconforming finite elements, SIAM J. Numer. Anal.,
39(2), 363-384, 2001.
@article{"International Journal of Architectural, Civil and Construction Sciences:63008", author = "Rado Flajs and Miran Saje", title = "Note on the Necessity of the Patch Test", abstract = "We present a simple nonconforming approximation of the linear two–point boundary value problem which violates patch test requirements. Nevertheless the solutions, obtained from these type of approximations, converge to the exact solution.
", keywords = "Generalized patch test, Irons' patch test, nonconforming finite element, convergence.", volume = "6", number = "9", pages = "741-4", }
