Convergence of $$\text{ dG(1) }$$ dG(1) in elastoplastic evolution

Abstract

The discontinuous Galerkin (dG) methodology provides a hierarchy of time discretization schemes for evolutionary problems such as elastoplasticity with the Prandtl-Reus flow rule. A dG time discretization has been proposed for a variational inequality in the context of rate-independent inelastic material behaviour in Alberty and Carstensen in (CMME 191:4949–4968, 2002) with the help of duality in convex analysis to justify certain jump terms. This paper establishes the first a priori error analysis for the dG(1) scheme with discontinuous piecewise linear polynomials in the temporal and lowest-order finite elements for the spatial discretization. Compared to a generalized mid-point rule, the dG(1) formulation distributes the action of the material law in the form of the variational inequality in time and so it introduces an error in the material law. This may result in a suboptimal convergence rate for the dG(1) scheme and this paper shows that the stress error in the $$L^\infty (L^2)$$ norm is merely $$O(h+k^{3/2})$$ based on a seemingly sharp error analysis. The numerical investigation for a benchmark problem with known analytic solution provides empirical evidence of a higher convergence rate of the dG(1) scheme compared to dG(0).