Least squares methods for elliptic systems
Authors:
A. K. Aziz, R. B. Kellogg and A. B. Stephens
Journal:
Math. Comp. 44 (1985), 53-70
MSC:
Primary 65N30; Secondary 76D07
DOI:
https://doi.org/10.1090/S0025-5718-1985-0771030-5
MathSciNet review:
771030
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Abstract | References | Similar Articles | Additional Information
Abstract: A weighted least squares method is given for the numerical solution of elliptic partial differential equations of Agmon-Douglis-Nirenberg type and an error analysis is provided. Some examples are given.
- S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35–92. MR 162050, DOI https://doi.org/10.1002/cpa.3160170104
- I. Babuška, J. T. Oden, and J. K. Lee, Mixed-hybrid finite element approximations of second-order elliptic boundary-value problems, Comput. Methods Appl. Mech. Engrg. 11 (1977), no. 2, 175–206. MR 451771, DOI https://doi.org/10.1016/0045-7825%2877%2990058-5
- Garth A. Baker, Simplified proofs of error estimates for the least squares method for Dirichlet’s problem, Math. Comp. 27 (1973), 229–235. MR 327056, DOI https://doi.org/10.1090/S0025-5718-1973-0327056-0
- James H. Bramble and Joachim A. Nitsche, A generalized Ritz-least-squares method for Dirichlet problems, SIAM J. Numer. Anal. 10 (1973), 81–93. MR 314284, DOI https://doi.org/10.1137/0710010
- James H. Bramble and Alfred H. Schatz, Rayleigh-Ritz-Galerkin methods for Dirichlet’s problem using subspaces without boundary conditions, Comm. Pure Appl. Math. 23 (1970), 653–675. MR 267788, DOI https://doi.org/10.1002/cpa.3160230408
- J. H. Bramble and A. H. Schatz, Least squares methods for $2m$th order elliptic boundary-value problems, Math. Comp. 25 (1971), 1–32. MR 295591, DOI https://doi.org/10.1090/S0025-5718-1971-0295591-8
- James H. Bramble and Ridgway Scott, Simultaneous approximation in scales of Banach spaces, Math. Comp. 32 (1978), no. 144, 947–954. MR 501990, DOI https://doi.org/10.1090/S0025-5718-1978-0501990-5
- J. H. Bramble and V. Thomée, Pointwise bounds for discrete Green’s functions, SIAM J. Numer. Anal. 6 (1969), 583–590. MR 263265, DOI https://doi.org/10.1137/0706053
- George J. Fix, Max D. Gunzburger, and R. A. Nicolaides, On finite element methods of the least squares type, Comput. Math. Appl. 5 (1979), no. 2, 87–98. MR 539567, DOI https://doi.org/10.1016/0898-1221%2879%2990062-2 G. J. Fix & E. Stephan, Finite Element Methods of the Least Squares Type for Regions With Corners, Report No. 81-41, December 16, 1981, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, Virginia 23665.
- Dennis C. Jespersen, A least squares decomposition method for solving elliptic equations, Math. Comp. 31 (1977), no. 140, 873–880. MR 461948, DOI https://doi.org/10.1090/S0025-5718-1977-0461948-0 J. L. Lions & E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. 1, Springer, Berlin, 1972. J. Roǐtberg & 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. 447-450. R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, New York, 1977.
- W. L. Wendland, Elliptic systems in the plane, Monographs and Studies in Mathematics, vol. 3, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. MR 518816
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Article copyright:
© Copyright 1985
American Mathematical Society