A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system

Abstract

In this paper we investigate a priori error estimates for the spacetime Galerkin finite element discretization of an optimal control problem governed by a simplified linear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of an elliptic PDE which has to be fulfilled at almost all times coupled with an ODE that has to hold true in almost all points in space. The state equation is discretized by a piecewise constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. For the discretization of the control we employ the same discretization technique which turns out to be equivalent to a variational discretization approach. We provide error estimates of optimal order both for the discretization of the state equation as well as for the optimal control. Numerical experiments are added to illustrate the proven rates of convergence.

A priorierror estimates for the space-time finite element approximation of a quasilinear gradient enhanced damage model

“…is identical to the fixed point equation in [19] (up to a different solution operator Φ), existence of a unique solution d τ,m ∈ L 2 (Ω) can be proven as in [19], Prop.…”

“…5.9, and [32], p. 105. Thus, we only prove the Lipschitz continuity (20). According to the definition of the partial derivative w.r.t.…”

“…Thus, we have a look at the regularity of ϕ = Φ(l, d) first. We assume u ∈ W 1,p D (Ω) and d ∈ H 1 (Ω) since according to [19], Thm. 3.12, the semidiscrete solution d is more regular in space.…”

“…We will use constant discontinuous finite elements for the discretization in time and usual H 1 -conforming linear finite elements for the discretization in space. For a first analysis of the discretization of such a model we simplified the underlying PDE system, skipping the displacement variable u as well as the nonlinear material function, see [19] for details. A further simplified linear version of this model, that lacks the max-operator in the ODE, has been studied in a companion paper, see [20].…”

“…For a first analysis of the discretization of such a model we simplified the underlying PDE system, skipping the displacement variable u as well as the nonlinear material function, see [19] for details. A further simplified linear version of this model, that lacks the max-operator in the ODE, has been studied in a companion paper, see [20]. The aim of this paper is to establish a priori discretization error estimates for the finite element discretization of (1)-(4).…”