Numerical Analysis of Elliptic Hemivariational Inequalities

Weimin Han (1) , Mircea Sofonea (2) , Mikaël Barboteu (2)

1 Department of Mathematics, University of Iowa
2 LAMPS - LAboratoire de Modélisation Pluridisciplinaire et Simulations

This paper is devoted to a study of the numerical solution of elliptic hemivariational inequalities with or without convex constraints by the finite element method. For a general family of elliptic hemivariational inequalities that facilitates error analysis for numerical solutions, the solution existence and uniqueness are proved. The Galerkin approximation of the general elliptic hemivariational inequality is shown to converge, and Céa's inequality is derived for error estimation. For various elliptic hemivariational inequalities arising in contact mechanics, we provide error estimates of their numerical solutions, which are of optimal order for the linear finite element method, under appropriate solution regularity assumptions. Numerical examples are reported on using linear elements to solve sample contact problems, and the simulation results are in good agreement with the theoretically predicted linear convergence.

Mots clés: elliptic hemivariational inequality; Clarke subdifferential; Galerkin approximation; finite element method; convergence; error estimates; contact problems

Domaines: Mathématiques [math]

Weimin Han, Mircea Sofonea, Mikaël Barboteu. Numerical Analysis of Elliptic Hemivariational Inequalities. SIAM Journal on Numerical Analysis, 2017, 55 (2), pp.640-663. ⟨10.1137/16M1072085⟩. ⟨hal-01626946⟩