Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Projection methods for Dirichlet's problem in approximating polygonal domains with boundary-value corrections


Authors: James H. Bramble, Todd Dupont and Vidar Thomée
Journal: Math. Comp. 26 (1972), 869-879
MSC: Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-1972-0343657-7
MathSciNet review: 0343657
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Consider Dirichlet's problem in a plane domain $ \Omega $ with smooth boundary $ \partial \Omega $. For the purpose of its approximate solution, an approximating domain $ {\Omega _h},0 < h \leqq 1$, with polygonal boundary $ \partial {\Omega _h}$ is introduced where the segments of $ \partial {\Omega _h}$ have length at most $ h$. A projection method introduced by Nitsche [6] is then applied on $ {\Omega _h}$ to give an approximate solution in a finite-dimensional subspace of functions $ {S_h}$, for instance a space of splines defined on a triangulation of $ {\Omega _h}$. The boundary terms in the bilinear form associated with Nitsche's method are modified to correct for the perturbation of the boundary.


References [Enhancements On Off] (What's this?)

  • [1] S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand Math. Studies, no. 2, Van Nostrand, Princeton, N. J., 1965. MR 31 #2504. MR 0178246 (31:2504)
  • [2] J. H. Bramble & V. Thomée, ``Semidiscrete Galerkin methods for parabolic problems,'' Math. Comp., v. 26, 1972, pp. 633-648. MR 0349038 (50:1532)
  • [3] J. H. Bramble & M. Zlámal, ``Triangular elements in the finite element method,'' Math. Comp., v. 24, 1970, pp. 809-820. MR 0282540 (43:8250)
  • [4] J. L. Lions & E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, no. 17, Dunod, Paris, 1968. MR 40 #512. MR 0247243 (40:512)
  • [5] J. Nitsche, ``Lineare Spline-Funktionen und die Methoden von Ritz für elliptische Randwertprobleme,'' Arch. Rational Mech. Anal., v. 36, 1970, pp. 348-355. MR 0255043 (40:8250)
  • [6] J. Nitsche, ``Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind,'' Abh. Math. Sem. Univ. Hamburg, v. 36, 1971, pp. 9-15. MR 0341903 (49:6649)
  • [7] G. Strang & A. E. Berger, ``The change in solution due to change in domain,'' (To appear.) MR 0337023 (49:1796)
  • [8] V. Thomée, ``Polygonal domain approximation in Dirichlet's problem,'' J. Inst. Math. Appl. (To appear.) MR 0349044 (50:1538)
  • [9] M. Zlámal, ``On the finite element method,'' Numer. Math., v. 12, 1968, pp. 394-409. MR 39 #5074. MR 0243753 (39:5074)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N30

Retrieve articles in all journals with MSC: 65N30


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1972-0343657-7
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society