Stabilized mixed hp-BEM for frictional contact problems in linear elasticity

Abstract

We investigate hp-stabilization for variational inequalities and boundary element methods based on the approach introduced by Barbosa and Hughes for finite elements. Convergence of a stabilized mixed boundary element method is shown for unilateral frictional contact problems for the Lamé equation. Without stabilization, the inf-sup constant need not be bounded away from zero for natural discretizations, even for fixed h and p. Both a priori and a posteriori error estimates are presented in the case of Tresca friction, for discretizations based on Bernstein or Gauss-Lobatto-Lagrange polynomials as test and trial functions. We also consider an extension of the a posteriori estimate to Coulomb friction. Numerical experiments underline our theoretical results.

Comparison of mixed hp-BEM (stabilized and non-stabilized) for frictional contact problems

“…However, its a priori error estimate is not yet fully clear due to the missing uniformity results on the discrete inf-sup condition. For the stabilized methods, which circumvents the discrete inf-sup condition, an a priori error estimate is proven in [13]. This stabilized method allows for an extremely flexible discretization of the Lagrange multiplier and seems to perform superior in terms of convergence rates for lowest order methods, uniform and adaptive h-versions.…”

“…affinely transformed Gauss-Lobatto points,T k is a boundary mesh of Γ C which may coincide with T h | Γ C , since we do not need a discrete inf-sup condition in the stabilized method, q is a polynomial degree distribution on that mesh and µ kq are linear combinations of Gauss-Lobatto-Lagrange (GLL) basis functions. Alternatively, one might use Bernstein polynomials (see [13]). Note that the ansatz functions in V hp can be written as linear combinations of GLL basis functions.…”

“…While mixed finite element approaches for contact problems are preferably done for low order elements [11] we focus here on recent results for the use of higher order BEM and especially on hp-adaptive procedures which we report from [12,13]. For approaches with variational inequalities (not mixed) see [8][9][10] whereas for obstacle problems we refer to [14][15][16].…”

“…Stabilized solution of the Coulomb-frictional problem, uniform mesh 256 elements, p = 1,[13]. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)…”

“…(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Error estimates for different families of discrete solutions (Coulomb-friction)[12,13].…”