Numerical Analysis of Elliptic Hemivariational Inequalities

Abstract

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.

Penalty and regularization method for variational‐hemivariational inequalities with application to frictional contact

“…We conclude that Theorem 23 provides the weak solvability of Problems 16 and 19, and the weak solutions satisfy the regularity ∈ , ∈ , , ∈ . The displacements and ∈ are determined uniquely, and, in general, , ∈  are not unique because of multivalued constitutive laws (12) and (30).…”

“…The proof is completed. □ A pair of functions ( , ) which satisfies (12) and Problem 18 is called a weak solution to Problem 16. A pair of functions ( , ) which satisfies (30) and Problem 22 is called a weak solution to Problem 19.…”

Convergence rate of multiscale finite element method for various boundary problems

“…The Céa's inequality for hemivariational problem is slightly different (cf. [11]), and we can obtain…”

Numerical studies of a hemivariational inequality for a viscoelastic contact problem with damage

“…Detailed discussion of the finite element method for solving HVIs can be found in [20]. More recently, there has been substantial progress in numerical analysis of HVIs, especially on optimal order error estimates for numerical solutions of HVIs, starting with the paper [14], followed by a sequence of papers, e.g., [2,18,13,19]; the reader is referred to [17] for a recent survey.…”