In this paper, a priori error estimates are derived for the mixed finite element discretization of optimal control problems governed by fourth order elliptic partial differential equations. The state and co-state are discretized by Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. The error estimates derived for the state variable as well as those for the control variable seem to be new. We illustrate with a numerical example to confirm our theoretical results.

In this paper, we propose a splitting positive definite mixed finite element method for the approximation of convex optimal control problems governed by linear parabolic equations, where the primal state variable y and its flux σ are approximated simultaneously. By using the first order necessary and sufficient optimality conditions for the optimization problem, we derive another pair of adjoint state variables z and ω, and also a variational inequality for the control variable u is derived. As we can see the two resulting systems for the unknown state variable y and its flux σ are splitting, and both symmetric and positive definite. Besides, the corresponding adjoint states z and ω are also decoupled, and they both lead to symmetric and positive definite linear systems. We give some a priori error estimates for the discretization of the states, adjoint states and control, where Ladyzhenkaya-Babuska-Brezzi consistency condition is not necessary for the approximation of the state variable y and its flux σ. Finally, numerical experiments are given to show the efficiency and reliability of the splitting positive definite mixed finite element method.

From Kantorovich’s theory we present a semilocal convergence result for Newton’s methodwhich is based mainly on a modification of the condition required to the second derivativeof the operator involved. In particular, instead of requiring that the second derivativeis bounded, we demand that it is centered. As a consequence, we obtain a modification ofthe starting points for Newton’s method. We illustrate this study with applications tononlinear integral equations of mixed Hammerstein type.

We examine an elliptic optimal control problem with control and state constraints inℝ3. An improved error estimate of 𝒪(hs) with 3/4 ≤ s ≤ 1 − ε is proven for a discretisationinvolving piecewise constant functions for the control and piecewise linear for the state.The derived order of convergence is illustrated by a numerical example.

In this paper, the finite element approximation of a class of semilinear parabolic optimal control problems with pointwise control constraint is studied. We discretize the state and co-state variables by piecewise linear continuous functions, and the control variable is approximated by piecewise constant functions or piecewise linear discontinuous functions. Some a priori error estimates are derived for both the control and state approximations. The convergence orders are also obtained.

In this paper, we investigate a priori error estimates for the quadratic optimal control problems governed by semilinear elliptic partial differential equations using higher order triangular mixed finite element methods. The state and the co-state are approximated by the order k Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k (k ≥ 0). A priori error estimates for the mixed finite element approximation of semilinear control problems are obtained. Finally, we present some numerical examples which confirm our theoretical results.

This paper presents the numerical analysis for a variational formulation of rate-independent phase transformations in elastic solids due to Mielke et al. The new model itself suggests an implicit time-discretization which is combined with thefinite element method in space. A priori error estimates are established for the quasioptimal spatial approximation of the stress fieldwithin one time-step. A posteriorierror estimates motivate an adaptive mesh-refining algorithm for efficient discretization. The proposed scheme enables numerical simulations which show that the model allows for hysteresis.