The finite element method with Lagrange multipliers for domains with corners

Pitkäranta, Juhani

doi:10.1090/S0025-5718-1981-0616357-9

The finite element method with Lagrange multipliers for domains with corners
HTML articles powered by AMS MathViewer

by Juhani Pitkäranta PDF

Math. Comp. 37 (1981), 13-30 Request permission

Abstract:

We study the convergence of the finite element method with Lagrange multipliers for approximately solving the Dirichlet problem for a second-order elliptic equation in a plane domain with piecewise smooth boundary. Assuming mesh refinements around the corners, we construct families of boundary subspaces that are compatible with triangular Lagrange elements in the interior, and we carry out the error analysis of the resulting approximations in weighted Sobolev spaces.

References

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957

Numer. Math.

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
I. Babuška, R. B. Kellogg, and J. Pitkäranta, Direct and inverse error estimates for finite elements with mesh refinements, Numer. Math. 33 (1979), no. 4, 447–471. MR 553353, DOI 10.1007/BF01399326

Interpolation Spaces

F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129–151 (English, with French summary). MR 365287
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
V. A. Kondrat′ev, Boundary value problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obšč. 16 (1967), 209–292 (Russian). MR 0226187
Juhani Pitkäranta, Boundary subspaces for the finite element method with Lagrange multipliers, Numer. Math. 33 (1979), no. 3, 273–289. MR 553590, DOI 10.1007/BF01398644
Juhani Pitkäranta, Local stability conditions for the Babuška method of Lagrange multipliers, Math. Comp. 35 (1980), no. 152, 1113–1129. MR 583490, DOI 10.1090/S0025-5718-1980-0583490-9
Hans Triebel, Interpolation theory for function spaces of Besov type defined in domains. II, Math. Nachr. 58 (1973), 63–86. MR 361760, DOI 10.1002/mana.19730580106