The Lagrange multiplier method for Dirichlet's problem

Author:
James H. Bramble

Journal:
Math. Comp. **37** (1981), 1-11

MSC:
Primary 65N30

DOI:
https://doi.org/10.1090/S0025-5718-1981-0616356-7

MathSciNet review:
616356

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Abstract: The Lagrange multiplier method of Babuška for the approximate solution of Dirichlet's problem for second order elliptic equations is reformulated. Based on this formulation, new estimates for the error in the solution and the boundary flux are given. Efficient methods for the solution of the approximate problem are discussed.

**[1]**Owe Axelsson,*Solution of linear systems of equations: iterative methods*, Sparse matrix techniques (Adv. Course, Technical Univ. Denmark, Copenhagen, 1976) Springer, Berlin, 1977, pp. 1–51. Lecture Notes in Math., Vol. 572. MR**0448834****[2]**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****[3]**Ivo Babuška,*The finite element method with Lagrangian multipliers*, Numer. Math.**20**(1972/73), 179–192. MR**0359352**, https://doi.org/10.1007/BF01436561**[4]**J. H. Bramble and J. E. Osborn,*Rate of convergence estimates for nonselfadjoint eigenvalue approximations*, Math. Comp.**27**(1973), 525–549. MR**0366029**, https://doi.org/10.1090/S0025-5718-1973-0366029-9**[5]**James H. Bramble and Ridgway Scott,*Simultaneous approximation in scales of Banach spaces*, Math. Comp.**32**(1978), no. 144, 947–954. MR**501990**, https://doi.org/10.1090/S0025-5718-1978-0501990-5**[6]**James W. Cooley and John W. Tukey,*An algorithm for the machine calculation of complex Fourier series*, Math. Comp.**19**(1965), 297–301. MR**0178586**, https://doi.org/10.1090/S0025-5718-1965-0178586-1**[7]**Richard S. Falk,*A Ritz method based on a complementary variational principle*, Rev. Française Automat. Informat. Recherche Opérationnelle Sér.**10**(1976), no. R-2, 39–48 (English, with Loose French summary). MR**0433915****[8]**J. L. Lions & E. Magenes,*Non-Homogeneous Boundary Value Problems and Applications*, Vol. 1, Springer-Verlag, Berlin and New York, 1972.**[9]**Martin Schechter,*On 𝐿^{𝑝} estimates and regularity. II*, Math. Scand.**13**(1963), 47–69. MR**0188616**, https://doi.org/10.7146/math.scand.a-10688

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DOI:
https://doi.org/10.1090/S0025-5718-1981-0616356-7

Article copyright:
© Copyright 1981
American Mathematical Society