Least squares methods for elliptic systems
Authors:
A. K. Aziz, R. B. Kellogg and A. B. Stephens
Journal:
Math. Comp. 44 (1985), 5370
MSC:
Primary 65N30; Secondary 76D07
MathSciNet review:
771030
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Abstract 
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Abstract: A weighted least squares method is given for the numerical solution of elliptic partial differential equations of AgmonDouglisNirenberg type and an error analysis is provided. Some examples are given.
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 S. Agmon, A. Douglis & L. Nirenberg, "Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II," Comm. Pure Appl. Math., v. .17, 1964, pp. 3592. MR 0162050 (28:5252)
 [2]
 I. Babuška, J. T. Oden & J. K. Lee, "Mixedhybrid finite element approximations of secondorder boundary value problems," Comput. Methods Appl. Mech. Engrg., v. 11, 1977, pp. 175206. MR 0451771 (56:10053)
 [3]
 G. A. Baker, "Simplified proofs of error estimates for the least squares method for Dirichlet's problem," Math. Comp., v. 27, 1973, pp. 229235. MR 0327056 (48:5398)
 [4]
 J. H. Bramble & J. A. Nitsche, "A generalized Ritzleastsquares method for Dirichlet problems," SIAM J. Numer. Anal., v. 10, 1973, pp. 8193. MR 0314284 (47:2836)
 [5]
 J. H. Bramble & A. H. Schatz, "RayleighRitzGalerkinmethods for Dirichlet's problem using subspaces without boundary conditions," Comm. Pure Appl. Math., v. 23, 1970, pp. 653675. MR 0267788 (42:2690)
 [6]
 J. H. Bramble & A. H. Schatz, "Least squares for 2mth order elliptic boundaryvalue problems," Math. Comp., v. 25, 1971, pp. 132. MR 0295591 (45:4657)
 [7]
 J. H. Bramble & R. Scott, "Simultaneous approximation in scales of Banach spaces," Math. Comp., v. 32, 1978, pp. 947954. MR 501990 (80a:65222)
 [8]
 J. H. Bramble & V. Thomée, "Pointwise bound for discrete Green's functions," SIAM J. Numer. Anal., v. 6, 1969, pp. 583590. MR 0263265 (41:7870)
 [9]
 G. J. Fix, M. D. Gunzburger, & R. A. Nicolaides, "On finite element methods of the least squares type," Comput. Math. Appl., v. 5, 1979, pp. 8798. MR 539567 (81b:65103)
 [10]
 G. J. Fix & E. Stephan, Finite Element Methods of the Least Squares Type for Regions With Corners, Report No. 8141, December 16, 1981, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, Virginia 23665.
 [11]
 D. C. Jesperson, "A least squares decomposition method for solving elliptic equations," Math. Comp., v. 31, 1977, pp. 873880. MR 0461948 (57:1930)
 [12]
 J. L. Lions & E. Magenes, NonHomogeneous Boundary Value Problems and Applications, Vol. 1, Springer, Berlin, 1972.
 [13]
 J. Roitberg & Z. Šeftel, "A theorem about the complete set of isomorphisms for systems elliptic in the sense of Douglis and Nirenberg," Ukrain. Mat. Zh., 1975, pp. 447450.
 [14]
 R. Temam, NavierStokes Equations, NorthHolland, Amsterdam, New York, 1977.
 [15]
 W. L. Wendland, Elliptic Systems in the Plane, Pitman, London, 1979. MR 518816 (80h:35053)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198507710305
PII:
S 00255718(1985)07710305
Article copyright:
© Copyright 1985
American Mathematical Society
