Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

A robust finite element method for nonhomogeneous Dirichlet problems in domains with curved boundaries


Authors: James H. Bramble and J. Thomas King
Journal: Math. Comp. 63 (1994), 1-17
MSC: Primary 65N30; Secondary 65F10
DOI: https://doi.org/10.1090/S0025-5718-1994-1242055-6
MathSciNet review: 1242055
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider a simple finite element method on an approximate polygonal domain using linear elements. The Dirichlet data are transferred in a natural way and the resulting linear system can be solved using multigrid techniques. Our analysis takes into account the change in domain and data transfer, and optimal-error estimates are obtained that are robust in the regularity of the boundary data provided they are at least square integrable. It is proved that the natural extension of our finite element approximation to the original domain is optimal-order accurate.


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

  • [1] A. Berger, $ {L_2}$ error estimates for finite elements with interpolated boundary conditions, Numer. Math. 21 (1973), 345-349. MR 0343655 (49:8395)
  • [2] A. Berger, L. R. Scott, and G. Strang, Approximate boundary conditions in the finite element method, Symposia Mathematica, X, Academic Press, New York, 1972, pp. 295-313. MR 0403258 (53:7070)
  • [3] J. J. Blair, Bounds for the change in the solutions of second order elliptic PDE's when the boundary is perturbed, SIAM J. Appl. Math. 24 (1973), 277-285. MR 0317557 (47:6104)
  • [4] J. H. Bramble, Multigrid methods, Lecture notes, Cornell University, 1992. MR 1247694 (95b:65002)
  • [5] J. H. Bramble, T. Dupont, and V. Thomée, Projection methods for Dirichlet's problem in approximating polygonal domains with boundary value corrections, Math. Comp. 26 (1972), 869-879. MR 0343657 (49:8397)
  • [6] J. H. Bramble and J. E. Pasciak, New estimates for multilevel algorithms including the V-cycle, Math. Comp. 60 (1993), 447-471. MR 1176705 (94a:65064)
  • [7] J. H. Bramble, J. Pasciak, and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring. I, Math. Comp. 47 (1986), 103-134. MR 842125 (87m:65174)
  • [8] J. H. Bramble, J. Pasciak, and J. Xu, The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms, Math. Comp. 56 (1991), 1-34. MR 1052086 (91h:65159)
  • [9] J. H. Bramble and J. Xu, Some estimates for a weighted $ {L^2}$ projection, Math. Comp. 56 (1991), 463-476. MR 1066830 (91k:65140)
  • [10] G. Choudury and I. Lasiecka, Optimal convergence rates for semidiscrete approximations of parabolic problems with nonsmooth boudary data, Numer. Funct. Anal. Optim. 12(4&5) (1991), 469-485. MR 1159921 (93e:65134)
  • [11] P. G. Ciarlet, Basic error estimates for elliptic problems, Handbook of Numerical Analysis, Vol. II (P. G. Ciarlet and J. L. Lions, eds.), North-Holland, Amsterdam, 1991, pp. 17-351. MR 1115237
  • [12] P. G. Ciarlet and P. A. Raviart, The combined effect of curved boundaries and numerical integration in isoparametric finite element methods, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, ed.), Academic Press, New York, 1972, pp. 409-474. MR 0421108 (54:9113)
  • [13] M. Dauge, Elliptic boundary value problems on corner domains : smoothness and asymptotics of solutions, Lecture Notes in Math., vol. 1341, Springer-Verlag, New York, 1988. MR 961439 (91a:35078)
  • [14] T. Dupont, $ {L_2}$ error estimates for projecting methods for parabolic equations in approximating domains, Mathematical Aspects of Finite Elements in Partial Differential Equations (C. de Boor, ed.), Academic Press, New York, 1974, pp. 313-352.
  • [15] G. J. Fix, M. D. Gunzburger, and J. S. Peterson, On finite element approximations of problems having inhomogeneous essential boundary conditions, Comput. Math. Appl. 9 (1983), 687-700. MR 726817 (85b:65102)
  • [16] D. A. French and J. T. King, Approximation of an elliptic control problem by the finite element method, Numer. Funct. Anal. Optim. 12 (1991), 299-314. MR 1143001 (92m:65144)
  • [17] -, Analysis of a robust finite element approximation for a parabolic equation with rough boundary data, Math. Comp. 60 (1993), 79-104. MR 1153163 (93d:65098)
  • [18] P. Grisvard, Elliptic problems in nonsmooth domains, Pitman, London, 1985. MR 775683 (86m:35044)
  • [19] I. Lasiecka, Galerkin approximation of abstract parabolic boundary value problems with rough boundary data--$ {L_p}$ theory, Math. Comp. 47 (1986), 55-75. MR 842123 (87i:65187)
  • [20] J. L. Lions and E. Magenes, Nonhomogeneous boundary value problems and applications. I, Springer-Verlag, Berlin, 1972.
  • [21] J. Nitsche, Lineare Spline-funktionen und die Methoden von Ritz für elliptische Randwertprobleme, Arch. Rational Mech. Anal. 36 (1970), 348-355. MR 0255043 (40:8250)
  • [22] L. R. Scott, Interpolated boundary conditions in the finite element method, SIAM J. Numer. Anal. 12 (1975), 404-427. MR 0386304 (52:7162)
  • [23] G. Strang, Variational crimes in the finite element method, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (A. K. Aziz, ed.), Academic Press, New York, 1972, pp. 689-710. MR 0413554 (54:1668)
  • [24] G. Strang and A. E. Berger, The change in solution due to change in domain, Partial Differential Equations (D. C. Spencer, ed.), Proc. Sympos. Pure Math., vol. 23, Amer. Math. Soc., Providence, RI, 1973, pp. 199-205. MR 0337023 (49:1796)
  • [25] V. Thomée, Polygonal domain approximation in Dirichlet's problem, J. Inst. Math. Appl. 11 (1973), 33-44. MR 0349044 (50:1538)
  • [26] -, Approximate solution of Dirichlet's problem using approximating polygonal domains, Topics in Numerical Analysis (J. J. H. Miller, ed.), Academic Press, New York, 1973, pp. 311-328.
  • [27] J. Xu, Theory of multilevel methods, Thesis, Cornell University, Ithaca, New York, 1989.
  • [28] A. Ženíšek, Nonhomogeneous boundary conditions and curved triangular finite elements, Apl. Mat. 26 (1981), 121-141. MR 612669 (82g:65057)
  • [29] M. Zlámal, Curved elements in the finite element method. I, SIAM J. Numer. Anal. 10 (1973), 229-240. MR 0395263 (52:16060)
  • [30] -, Curved elements in the finite element method. II, SIAM J. Numer. Anal. 11 (1974), 347-362. MR 0343660 (49:8400)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N30, 65F10

Retrieve articles in all journals with MSC: 65N30, 65F10


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1994-1242055-6
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society