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Analysis of a robust finite element approximation for a parabolic equation with rough boundary data


Authors: Donald A. French and J. Thomas King
Journal: Math. Comp. 60 (1993), 79-104
MSC: Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-1993-1153163-1
MathSciNet review: 1153163
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Abstract: The approximation of parabolic equations with nonhomogeneous Dirichlet boundary data by a numerical method that consists of finite elements for the space discretization and the backward Euler time discretization is studied. The boundary values are assumed in a least squares sense. It is shown that this method achieves an optimal rate of convergence for rough (only $ {L^2}$) boundary data and for smooth data as well. The results of numerical computations which confirm the robust theoretical error estimates are also presented.


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  • [1] I. Babuška and J. Osborn, Eigenvalue problems, Handbook of Numerical Analysis, vol. II (P. G. Ciarlet and J. L. Lions, eds.), North-Holland, Amsterdam, 1991, pp. 641-787. MR 1115240
  • [2] R. Bank and T. Dupont, An optimal order process for solving finite element equations, Math. Comp. 36 (1981), 35-51. MR 595040 (82b:65113)
  • [3] 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)
  • [4] J. H. Bramble and V. Thomée, Discrete time Galerkin methods for a parabolic boundary value problem, Ann. Mat. Pura Appl. 101 (1974), 115-152. MR 0388805 (52:9639)
  • [5] P. L. Butzer and H. Berens, Semi-groups of operators and approximation, Springer-Verlag, Berlin, 1967. MR 0230022 (37:5588)
  • [6] G. Choudury, Fully discrete Galerkin approximations of parabolic boundary-value problems with nonsmooth boundary data, Numer. Math. 57 (1990), 179-203. MR 1048311 (91f:65165)
  • [7] P. G. Ciarlet, The finite element method for elliptic problems, North-Holland, Amsterdam, 1978. MR 0520174 (58:25001)
  • [8] K. Eriksson and C. Johnson, Error estimates and automatic time step control for nonlinear parabolic problems. I, SIAM J. Numer. Anal. 24 (1987), 12-23. MR 874731 (88e:65114)
  • [9] K. Eriksson, C. Johnson, and V. Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Modél. Math. Anal. Numér. 19 (1985), 611-643. MR 826227 (87e:65073)
  • [10] 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)
  • [11] D. 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)
  • [12] P. Grisvard, Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, Numerical Solution of Partial Differential Equations. III (B. Hubbard, ed.), Academic Press, New York, 1976, 207-274. MR 0466912 (57:6786)
  • [13] P. Grisvard, Elliptic problems in nonsmooth domains, Pitman, Boston, 1985. MR 775683 (86m:35044)
  • [14] C. Johnson, Numerical solutions of partial differential equations by the finite element method, Cambridge Univ. Press, Cambridge, 1987.
  • [15] C. Johnson, Y.-Y. Nie, and V. Thomée, An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem, SIAM J. Numer. Anal. 27 (1990), 277-291. MR 1043607 (91g:65199)
  • [16] G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control. Optim. 20 (1982), 414-427. MR 652217 (84h:49059)
  • [17] I. Lasiecka, Convergence estimates for semidiscrete approximations of nonselfadjoint parabolic equations, SIAM J. Numer. Anal. 21 (1984), 894-909. MR 760624 (85m:65100)
  • [18] -, Galerkin approximations of abstract parabolic boundary value problems with rough boundary data--$ {L_p}$ theory, Math. Comp. 47 (1986), 55-75. MR 842123 (87i:65187)
  • [19] -, Unified theory for abstract parabolic boundary problems--A semigroup approach, Appl. Math. Optim. 6 (1980), 287-333. MR 587501 (81m:35077)
  • [20] J. L. Lions and E. Magenes, Nonhomogeneous boundary value problems and applications. 1, 2, Springer-Verlag, Berlin, 1972.
  • [21] V. Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Math., vol. 1054, Springer-Verlag, Berlin, 1984. MR 744045 (86k:65006)
  • [22] R. Winther, Error estimates for a Galerkin approximation for a parabolic control problem, Ann. Mat. Pura Appl. 104 (1978), 173-206. MR 515960 (80a:49067)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1993-1153163-1
Keywords: Finite elements, parabolic equations, backward Euler method
Article copyright: © Copyright 1993 American Mathematical Society

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