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

French, Donald A.; King, J. Thomas

doi:10.1090/S0025-5718-1993-1153163-1

Analysis of a robust finite element approximation for a parabolic equation with rough boundary data
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by Donald A. French and J. Thomas King PDF

Math. Comp. 60 (1993), 79-104 Request permission

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.

References

I. Babuška and J. Osborn, Eigenvalue problems, Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 641–787. MR 1115240
Randolph E. Bank and Todd Dupont, An optimal order process for solving finite element equations, Math. Comp. 36 (1981), no. 153, 35–51. MR 595040, DOI 10.1090/S0025-5718-1981-0595040-2
J. H. Bramble, J. E. Pasciak, and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring. I, Math. Comp. 47 (1986), no. 175, 103–134. MR 842125, DOI 10.1090/S0025-5718-1986-0842125-3
James H. Bramble and Vidar Thomée, Discrete time Galerkin methods for a parabolic boundary value problem, Ann. Mat. Pura Appl. (4) 101 (1974), 115–152. MR 388805, DOI 10.1007/BF02417101
Paul L. Butzer and Hubert Berens, Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften, Band 145, Springer-Verlag New York, Inc., New York, 1967. MR 0230022
G. Choudury, Fully discrete Galerkin approximations of parabolic boundary-value problems with nonsmooth boundary data, Numer. Math. 57 (1990), no. 2, 179–203. MR 1048311, DOI 10.1007/BF01386406
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
Kenneth Eriksson and Claes Johnson, Error estimates and automatic time step control for nonlinear parabolic problems. I, SIAM J. Numer. Anal. 24 (1987), no. 1, 12–23. MR 874731, DOI 10.1137/0724002
Kenneth Eriksson, Claes Johnson, and Vidar Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Modél. Math. Anal. Numér. 19 (1985), no. 4, 611–643 (English, with French summary). MR 826227, DOI 10.1051/m2an/1985190406111
George J. Fix, Max D. Gunzburger, and Janet S. Peterson, On finite element approximations of problems having inhomogeneous essential boundary conditions, Comput. Math. Appl. 9 (1983), no. 5, 687–700. MR 726817, DOI 10.1016/0898-1221(83)90126-8
Donald A. French and J. Thomas King, Approximation of an elliptic control problem by the finite element method, Numer. Funct. Anal. Optim. 12 (1991), no. 3-4, 299–314. MR 1143001, DOI 10.1080/01630569108816430
Pierre Grisvard, Behavior of the solutions of an elliptic boundary value problem in a polygonal or polyhedral domain, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 207–274. MR 0466912
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683

Numerical solutions of partial differential equations by the finite element method

Claes Johnson, Yi Yong Nie, and Vidar 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), no. 2, 277–291. MR 1043607, DOI 10.1137/0727019
Greg Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim. 20 (1982), no. 3, 414–427. MR 652217, DOI 10.1137/0320032
Irena Lasiecka, Convergence estimates for semidiscrete approximations of nonselfadjoint parabolic equations, SIAM J. Numer. Anal. 21 (1984), no. 5, 894–909. MR 760624, DOI 10.1137/0721058
I. Lasiecka, Galerkin approximations of abstract parabolic boundary value problems with rough boundary data—$L_p$ theory, Math. Comp. 47 (1986), no. 175, 55–75. MR 842123, DOI 10.1090/S0025-5718-1986-0842123-X
Irena Lasiecka, Unified theory for abstract parabolic boundary problems—a semigroup approach, Appl. Math. Optim. 6 (1980), no. 4, 287–333. MR 587501, DOI 10.1007/BF01442900

Nonhomogeneous boundary value problems and applications

Vidar Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Mathematics, vol. 1054, Springer-Verlag, Berlin, 1984. MR 744045
Ragnar Winther, Error estimates for a Galerkin approximation of a parabolic control problem, Ann. Mat. Pura Appl. (4) 117 (1978), 173–206. MR 515960, DOI 10.1007/BF02417890