VERIFIED CALCULATION OF AN ERROR BOUND FOR A SIMPLIFIED ELASTOPLASTICITY MODEL

J ROHE

uafx@rz.uni-karlsruhe.de

Mathematisches Institut I
Universität Karlsruhe
D-76128 Karlsruhe, Germany

We consider a simplified model for nonlinear plasticity. Let $f \in L^2(\Omega)$,

$$S_f = \{ \sigma \in (L_2(\Omega))^2 \vert - \textrm{div} \,\sigma = f \}$$

the equilibrium set, and

$$K_f = \{ \sigma \in S_f \vert \ \Vert\sigma\Vert _\infty \le k \},\ k > 0.$$

Then, the convex minimisation problem to be considered is

$$\min_{\sigma \in K_f} \Vert \sigma \Vert^2_{(L_2(\Omega))^2}$$

We will propose a method for enclosing the solution $\sigma$ when $\Omega$ ist a bounded, simply connected, polygonal domain in $\mathbf{R}^2$. If $K_f \neq \emptyset,$ there exists a unique solution. If an approximation for it in $K_f$ is known, an error bound can be given in the $L^2$ norm using duality methods. The main difficulty is the construction of an initial element in $K_f.$