Volume 17, Issue 3

This work is devoted to the discretisation of non linear elliptic problems on general polyhedral meshes in several space dimensions. The SUSHI scheme which was recently studied for anisotropic heterogeneous problems is applied in its full barycentric version, thus resulting into a cell centred scheme written under variational form, also known as ‘SUCCES’. We prove the existence of the approximate solution and its convergence to the weak solution of the continuous solution as the mesh size tends to 0. Numerical examples are shown for the p -Laplacian.

We devise an indirect multiple shooting (MS) method for PDE constrained optimization with a posteriori error estimates and local mesh refinement. MS is applied to the KKT system as is exemplified for a linear quadratic optimal control problem. The setup of MS in function spaces and their discrete analogs is discussed. Error representation formulas based on Galerkin orthogonality are derived. They involve sensitivity analysis by an adjoint problem and employ standard error representation on subintervals combined with additional projection errors at the shooting nodes. A posteriori error estimates and mesh refinement indicators are derived from this error representation. We discuss several mesh structures originating from different restrictions to local refinement. Finally, we present numerical results for the heat equation and problems exhibiting simple dynamics. In a second article, we will address the embedding of the scheme into a nonlinear shooting method.

We provide an a posteriori error analysis of finite element approximations of pointwise state constrained distributed optimal control problems for second order elliptic boundary value problems. In particular, we derive a residual-type a posteriori error estimator and prove its efficiency and reliability up to oscillations in the data of the problem and a consistency error term. In contrast to the case of pointwise control constraints, the analysis is more complicated, since the multipliers associated with the state constraints live in measure spaces. The analysis essentially makes use of appropriate regularizations of the multipliers both in the continuous and in the discrete regime. Numerical examples are given to illustrate the performance of the error estimator.