Abstract
In this paper, we present an a posteriori error analysis for the finite element approximation of a variational inequality. We derive a posteriori error estimators of residual type, which are shown to provide upper bounds on the discretization error for a class of variational inequalities provided the solutions are sufficiently regular. Furthermore we derive sharp a posteriori error estimators with both lower and upper error bounds for a subclass of the obstacle problem which are frequently met in many physical models. For sufficiently regular solutions, these estimates are shown to be equivalent to the discretization error in an energy type norm. Our numerical tests show that these sharp error estimators are both reliable and efficient in guiding mesh adaptivity for computing the free boundaries.
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REFERENCES
Ainsworth, M., Oden, J. T., and Lee, C. Y. (1993). Local a posteriori error estimators for variational inequalities. Numerical Methods for Partial Differential Equations 9, 22–33.
Ainsworth, M., and Oden, J. T. (1997). A posteriori error estimators in finite element analysis. Computer Methods Appl. Mech. Engr. 142, 1–88.
Bank, R. E., and Weiser, A. (1985). Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44, 283–301.
Baranger, J., and Amri, H. E. (1991). A Posteriori Error Estimators in Finite Element Approximation of Quasi-Newtonian Flows, M 2 AN, Vol. 25, pp. 31–48.
Bernardi, C. (1989). Optimal finite-element interpolation on curved domains. SIAM J. of Numer. Anal. 26, 1212–1240.
Brezis, H. (1972). Problè mes unilaté raux. J. Math. Pures. Appl. 51, 1–168.
Brezzi, F., Hager, W. W., and Raviart, P. A. (1977). Error estimates for the finite element solution of variational inequalities I. Numer. Math. 28, 431–443.
Ciarlet, P. G. (1978). The Finite Element Method for Elliptic Problems, North-Holland Publ., Amsterdam.
Clè ment, Ph. (1975). Approximation by finite element functions using local regularization. RAIRO Numer. Anal. R-2, 77–84.
Duvaut, G., and Lions, J. L. (1973). The Inequalities in Mechanics and Physics, Springer-Verlag.
Elliott, C. M. (1980). On a variational inequality formulation of an electrochemical machining and its approximation by finite element methods. J. Inst. Math. Appl. 25, 121–131.
Elliott, C. M., and Ockendon, J. R. (1982). Weak and Variational Methods for Moving Boundary Problems, Research Notes in Mathematics, Vol. 59, Pitman, Boston.
Falk, R. S. (1974). Error estimates for the approximation of a class of variational inequalities. Math. Comp. 28, 963–971.
Friedman, A. (1982). Variational Principles and Free-Boundary Problems, Academic Press, New York.
French, D. A., Larsson, S., and Nochetto, R. H. Pointwise A Posteriori Error Analysis for an Adaptive Penalty Finite Element Method for the Obstacle Problem (in preparation).
Glowinski, R., Lions, J. L., and Tremolieres, R. (1976). Numerical Analysis of Variational Inequalities, North-Holland, Netherlands.
Johnson, C. (1992). Adaptive finite element methods for the obstacle problem. Math. Models Methods Appl. Sci. 2, 483–487.
Kinderlehrer, D., and Stampacchia, G. (1980). An Introduction to Variational Inequalities and Their Applications, Academic Press, New York.
Kornhuber, R. (1996). A posteriori error estimates for elliptic variational inequalities. Comp. Math. Appl. 31, 49–60.
Kufner, A., John, O., and Fucik, S. (1977). Function Spaces, Nordhoff, Leyden, The Netherlands.
Lewy, H., and Stampacchia, G. (1969). On the regularity of the solution of a variational inequality. Comm. Pure Appl. Math. 22, 153–188.
Lions, J. L., and Magenes, E. (1972). Nonhomogeneous Boundary Value Problems and Applications (I), Springer-Verlag.
Liu, W. B., and Barrett, J. W. (1993). Error bounds for the finite element approximation of a degenerate quasilinear parabolic variational inequality. Adv. Comp. Math. 1(2), 223–239.
Liu, W. B., and Barrett, J. W. (1994). Quasi-norm error bounds for the finite element approximation of degenerate quasilinear elliptic variational inequalities. RAIRO Numer. Anal. 28, 725–744.
Liu, W. B., and Barrett, J. W. (1995). Quasi-norm error bounds for the finite element approximation of degenerate quasilinear parabolic variational inequalities. Num. Funct. Anal. Optim. 16, 1309–1321.
Nochetto. R. H. (1986). A note on the approximation of free boundaries by finite element methods. RAIRO Model. Math. Anal. Numer. 20, 355–368.
Nochetto, R. H. (1989). Pointwise accuracy of a finite element method for nonlinear variational inequalities. Numer. Math. 54, 601–618.
Scholz, R. (1986). Numerical solution of the obstacle problem by the penalty method, II. Numer. Math. 49, 255–268.
Scott, L. R., and Zhang, S. (1990). Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54, 483–493.
Verfurth, R. (1989). A posteriori error estimators for the Stokes equations. Numer. Math. 55, 309–325.
Verfurth, R. (1996). A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner.
Verfurth, R. (1994). A posteriori error estimates for non-linear problems. Math. Comp. 62, 445–475.
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Liu, W., Yan, N. A Posteriori Error Estimators for a Class of Variational Inequalities. Journal of Scientific Computing 15, 361–393 (2000). https://doi.org/10.1023/A:1011130501691
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DOI: https://doi.org/10.1023/A:1011130501691