Abstract
In this paper we present a priori error estimates for the Galerkin solution of variational inequalities which are formulated in fractional Sobolev trace spaces, i.e. in \(\widetilde{H}^{1/2}(\Gamma )\). In addition to error estimates in the energy norm we also provide, by applying the Aubin–Nitsche trick for variational inequalities, error estimates in lower order Sobolev spaces including \(L_2(\Gamma )\). The resulting discrete variational inequality is solved by using a semi-smooth Newton method, which is equivalent to an active set strategy. A numerical example is given which confirms the theoretical results.
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References
Brenner, S., Scott, R.L.: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)
Brezis, H.: Problémes unilateraux. J. Math. Pures Appl. 51, 1–168 (1972)
Brezzi, F., Hager, W.W., Raviart, P.A.: Error estimates for the finite element solution of variational inequalities. Numer. Math. 28, 431–443 (1977)
Carstensen, C., Gwinner, J.: FEM and BEM coupling for a nonlinear transmission problem with Signorini contact. SIAM J. Numer. Anal. 34, 1845–1864 (1997)
Chen, X., Nashed, Z., Qi, L.: Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J. Numer. Anal. 38, 1200–1216 (2001)
Clement, P.: Approximation by finite element functions using local regularization. RAIRO Anal. Numer. R–2, 77–84 (1975)
Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19, 613–626 (1988)
Eck, C., Jarušek, J., Krbec, M.: Unilateral contact problems. Variational methods and existence theorems. Pure Appl. Math. 270. Chapman & Hall/CRC, Boca Raton (2005)
Eck, C., Steinbach, O., Wendland, W.L.: A symmetric boundary element method for contact problems with friction. Math. Comput. Simul. 50, 43–61 (1999)
Falk, R.S.: Error estimates for the approximation of a class of variational inequalities. Math. Comput. 28, 963–971 (1974)
Glowinski, R.: Numerical methods for nonlinear variational problems. Springer, Berlin (1980)
Gwinner, J., Stephan, E.P.: A boundary element procedure for contact problems in plane linear elastostatics. RAIRO Model. Math. Anal. Numer. 27, 457–480 (1993)
Han, H.: A direct boundary element method for Signorini problems. Math. Comp. 55, 115–128 (1990)
Hintermüller, M., Ito, K., Kunisch, K.: The primal-dual active set strategy as a semi-smooth Newton method. SIAM J. Optim. 13, 865–888 (2002)
Hintermüller, M., Ulbrich, M.: A mesh-independence result for semi-smooth Newton methods. Math. Program. Ser. B 101, 151–184 (2004)
Hsiao, G.C., Wendland, W.L.: The Aubin–Nitsche lemma for integral equations. J. Integral Equ. 3, 299–315 (1981)
Hüeber, S., Wohlmuth, B.: An optimal a priori error estimate for nonlinear multibody contact problems. SIAM J. Numer. Anal. 43, 156–173 (2005)
Isac, G.: Complementarity Problems. Lecture Notes in Mathematics, vol. 1528. Springer, Berlin (1992)
Ito, K., Kunisch, K.: Semi-smooth Newton methods for the Signorini problem. Appl. Math. 53, 455–468 (2008)
Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)
Lions, J.L., Stampacchia, G.: Variational inequalities. Commun. Pure Appl. Math. 20, 493–519 (1967)
Maischak, M., Stephan, E.P.: Adaptive hp-versions of BEM for Signorini problems. Appl. Numer. Math. 54, 425–449 (2005)
McLean, W., Steinbach, O.: Boundary element preconditioners for a hypersingular integral equation on an interval. Adv. Comput. Math. 11, 271–286 (1999)
Natterer, F.: Optimale \(L_2\)-Konvergenz finiter Elemente bei Variationsungleichungen. Bonn. Math. Schrift. 89, 1–12 (1976)
Of, G., Phan, T.X., Steinbach, O.: Boundary element methods for Dirichlet boundary control problems. Math. Methods Appl. Sci. 33, 2187–2205 (2010)
Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54, 483–493 (1990)
Spann, W.: On the boundary element method for the Signorini problem of the Laplacian. Numer. Math. 65, 337–356 (1993)
Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems. Finite and Boundary Elements. Springer, New York (2008)
Stephan, E.P., Wendland, W.L.: A hypersingular boundary integral method for two-dimensional screen and crack problems. Arch. Ration. Mech. Anal. 112, 363–390 (1990)
Suttmeier, F.-T.: Numerical solution of variational inequalities by adaptive finite elements. Vieweg + Teubner, Wiesbaden (2008)
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The author wants to thank the referee whose critical remarks and helpful hints and advises resulted in an essential improvement of the paper.
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In memoriam Christof Eck (1968–2011).