Mixed finite-element approximations of linear boundary-value problems

Authors:
J. N. Reddy and J. T. Oden

Journal:
Quart. Appl. Math. **33** (1975), 255-280

MSC:
Primary 65N30

DOI:
https://doi.org/10.1090/qam/451782

MathSciNet review:
451782

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A theory of mixed finite-element/Galerkin approximations of a class of linear boundary-value problems of the type is presented, in which appropriate notions of consistency, stability, and convergence are derived. Some error estimates are given and the results of a number of numerical experiments are discussed.

**[1]**L. R. Herrmann,*A bending analysis of plates, Proceedings of the Conference on Matrix Methods in Structural Mechanics*, Wright-Patterson AFB, Ohio, AFFDL-TR66-80, pp. 577-604, 1966**[2]**K. Hellan,*Analysis of elastic plates in flexure by a simplified finite element method*, Acta Polytechnica Scandinavica, Civil Engineering Series No. 46, Trondheim, 1967**[3]**W. Prager,*Variational principles for elastic plates with relaxed continuity requirements*, Int. J. Solids Structures**4**, 837-844 (1968)**[4]**W. Visser,*A refined mixed type plate bending element*, AIAA J.**7**, 1801-1803 (1969)**[5]**R. S. Dunham and K. S. Pister,*A finite element application of the Hellinger-Resissner variational theorem, in Proceedings of the Conference on Matrix Methods in Structural Mechanics*, Wright-Patterson AFB, Ohio, AFFDL-TR68-150, pp. 471-487, 1968**[6]**J. Bäcklund,*Mixed finite element analysis of plates in bending*, Chalmers Tekniska Hogskola Institutionen for byggnadsstatik Publication 71.4, Goteburg, 1972**[7]**W. Wünderlich,*Discretisation of structural problems by a generalized variational approach*, Papers presented at International Association for Shell Structures, Pacific Symposium on Hydrodynamically Loaded Shells-Part I, Honolulu, Hawaii, Oct. 10-15, 1971**[8]**T. H. H. Pain,*Formulations of finite element methods for solid continua*, in*Recent advances in matrix methods of structural analysis and Design*, R. H. Gallagher, Y. Yamada, and J. T. Oden (eds.), University of Alabama Press, University, 1971**[9]**P. Tong,*New Displacement hybrid finite element model for solid continua*, Int. J. Numer. Meth. Eng.**2**, 73-83 (1970)**[10]**T. H. H. Pian and P. Tong,*Basis of finite element methods for solid Continua*, Int. J. Numer. Meth. Eng.**1**, 3-28 (1969)**[11]**J. N. Reddy,*Accuracy and convergence of mixed finite-element approximations of thin bars, membranes, and plates on elastic foundations, in Proceedings of the Graduate Research Conference in Applied Mechanics*, Las Cruces, New Mexico, paper 1B5, March 1973**[12]**Claes Johnson,*On the convergence of a mixed finite-element method for plate bending problems*, Numer. Math.**21**(1973), 43–62. MR**0388807**, https://doi.org/10.1007/BF01436186**[13]**F. Kikuchi and Y. Ando,*On the convergence of a mixed finite element scheme for plate bending*, Nucl. Eng. Design 24, 357-373 (1973)**[14]**J. T. Oden,*Some contributions to the mathematical theory of mixed, finite element approximations*, in Tokyo Seminar on Finite Elements, Tokyo, Japan, The University of Tokyo press, 1973**[15]**J. T. Oden and J. N. Reddy,*On dual-complementary variational principles in mathematical physics*, Internat. J. Engrg. Sci.**12**(1974), 1–29 (English, with French, German, Italian and Russian summaries). MR**0451158**, https://doi.org/10.1016/0020-7225(74)90073-1**[16]**J. N. Reddy, and J. T. Oden,*Convergence of mixed finite element approximations of a class of linear Boundary-Value problems*, Struct. Mech.**2**, 83-108 (1973)**[17]**J. T. Oden,*Finite elements of nonlinear continua*, McGraw-Hill, New York, 1972**[18]**S. W. Key,*A convergence investigation of the direct stiffness method*, Doctoral Dissertation, University of Washington, Seattle, 1966**[19]**R. W. McLay,*Completeness and convergence properties of finite-element displacement functions-- a general treatment*, AIAA 5th Aerospace Science Meeting AIAA Paper 67-143, New York, 1967**[20]**M. W. Johnson, and R. W. McLay,*Convergence of the finite element method in the theory of elasticity*, J. Appl. Mech. E**35**, 274-278 (1968)**[21]**Ivo Babuška and A. K. Aziz,*Survey lectures on the mathematical foundations of the finite element method*, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR**0421106****[22]**Jean-Pierre Aubin,*Approximation of elliptic boundary-value problems*, Wiley-Interscience [A division of John Wiley & Sons, Inc.], New York-London-Sydney, 1972. Pure and Applied Mathematics, Vol. XXVI. MR**0478662****[23]**Martin H. Schultz,*Spline analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. Prentice-Hall Series in Automatic Computation. MR**0362832****[24]**Gilbert Strang and George J. Fix,*An analysis of the finite element method*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1973. Prentice-Hall Series in Automatic Computation. MR**0443377****[25]**Eugene Isaacson and Herbert Bishop Keller,*Analysis of numerical methods*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0201039****[26]**Arch W. Naylor and George R. Sell,*Linear operator theory in engineering and science*, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1971. MR**0461166****[27]**J. N. Reddy,*A mathematical theory of complementary-dual variational principles and mixed finite-element approximations of linear boundary-value problems in continuum mechanics*, Doctoral Dissertation, University of Alabama in Huntsville, May, 1974**[28]**P. G. Ciarlet and P.-A. Raviart,*General Lagrange and Hermite interpolation in 𝑅ⁿ with applications to finite element methods*, Arch. Rational Mech. Anal.**46**(1972), 177–199. MR**0336957**, https://doi.org/10.1007/BF00252458**[29]**P. G. Ciarlet and P.-A. Raviart,*Interpolation theory over curved elements, with applications to finite element methods*, Comput. Methods Appl. Mech. Engrg.**1**(1972), 217–249. MR**0375801**, https://doi.org/10.1016/0045-7825(72)90006-0**[30]**L. R. Herrmann, ``Finite-Element Bending Analysis for Plates,'' J. Eng. Mech. Div. ASCE**93**, 13-26 (1967)**[31]**J. J. Connor,*Mixed models for plates, in Proceedings of a Seminar on Finite-Element Techniques in Structural Mechanics*, University of Southampton, pp. 125-151, 1970

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
65N30

Retrieve articles in all journals with MSC: 65N30

Additional Information

DOI:
https://doi.org/10.1090/qam/451782

Article copyright:
© Copyright 1975
American Mathematical Society