On the stability and convergence of higherorder mixed finite element methods for secondorder elliptic problems
Author:
Manil Suri
Journal:
Math. Comp. 54 (1990), 119
MSC:
Primary 65N30; Secondary 65N10
MathSciNet review:
990603
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Abstract: We investigate the use of higherorder mixed methods for secondorder elliptic problems by establishing refined stability and convergence estimates which take into account both the mesh size h and polynomial degree p. Our estimates yield asymptotic convergence rates for the p and h  pversions of the finite element method. They also describe more accurately than previously proved estimates the increased rate of convergence expected when the hversion is used with higherorder polynomials. For our analysis, we choose the RaviartThomas and the BrezziDouglasMarini elements and establish optimal rates of convergence in both h and p (up to arbitrary ).
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 I. Babuška, The p and h  p versions of the finite element method. The state of the art, in Finite Elements Theory and Application (D. L. Dwoyer, M. Y. Hussaini and R. G. Voigt, eds.), SpringerVerlag, New York, 1988, pp. 199239. MR 964487 (90b:65197)
 [2]
 , Are high degree elements preferable? Some aspects of the h and h  p versions of the finite element method, in Numerical Techniques for Engineering Analysis and Design, Vol. I (G. N. Pande and J. Middleton, eds.), Martinus Nijhoff Publishers, 1987.
 [3]
 I. Babuška and M. R. Dorr, Error estimates for the combined h and p version of the finite element method, Numer. Math. 37 (1981), 257277. MR 623044 (82h:65080)
 [4]
 I. Babuška and T. Scapolla, The computational aspects of the h, p and h  p versions of the finite element method, in Advances in Computer Methods for PDEsVI (R. Vichnevetsky and R. S. Stepleman, eds.), IMACS, 1987.
 [5]
 I. Babuška and M. Suri, The h  p version of the finite element method with quasiuniform meshes, RAIRO Modél. Math. Anal. Numér. 21 (1987), 199238. MR 896241 (88d:65154)
 [6]
 , The treatment of nonhomogeneous Dirichlet boundary conditions by the pversion of the finite element method, Numer. Math. 55 (1989), 97121. MR 987158 (90m:65191)
 [7]
 I. Babuška and B. A. Szabo, Lecture notes on finite element analysis, In preparation.
 [8]
 I. Babuška, B. A. Szabo and I. N. Katz, The pversion of the finite element method, SIAM J. Numer. Anal. 18 (1981), 515545. MR 615529 (82j:65081)
 [9]
 F. Brezzi, J. Douglas and L. D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985), 217235. MR 799685 (87g:65133)
 [10]
 P. G. Ciarlet, The finite element method for elliptic problems, NorthHolland, Amsterdam, 1978. MR 0520174 (58:25001)
 [11]
 S. Jensen and M. Vogelius, Divergence stability in connection with the pversion of the finite element method, to appear in SIAM J. Numer. Anal., 1990. MR 1080717 (91m:65261)
 [12]
 C. Johnson and V. Thomée, Error estimates for some mixed finite element methods for parabolic type problems, RAIRO Anal. Numér. 15 (1981), 4178. MR 610597 (83c:65239)
 [13]
 J. L. Lions and E. Magenes, Nonhomogeneous boundary value problems and applications I, SpringerVerlag, Berlin and New York, 1972.
 [14]
 P. A. Raviart and J. M. Thomas, A mixed finite element method for second order elliptic problems, Proc. Sympos. Mathematical Aspects of the Finite Element Method (Rome, 1975), Lecture Notes in Math., vol. 606, SpringerVerlag, Berlin, 1977, pp. 292315. MR 0483555 (58:3547)
 [15]
 L. R. Scott and M. Vogelius, Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials, RAIRO Modél. Math. Anal. Numér. 19 (1985), 111143. MR 813691 (87i:65190)
 [16]
 E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1970. MR 0290095 (44:7280)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0025571819900990603X
PII:
S 00255718(1990)0990603X
Article copyright:
© Copyright 1990
American Mathematical Society
