Goal-oriented anisotropic mesh adaptation methods for nonlinear boundary value problems

  • Dolejsi, Vit (Charles University, Prague)

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We deal with the development of goal-oriented error estimates and adaptive methods for the numerical solution of nonlinear time-independent problems. In order to estimate a quantity of interest given by a target functional, an adjoint problem has to be formulated and solved, cf. [1, 4]. The adjoint problem is usually introduced by a differentiation of the primal weak formulation. However, the differentiation is problematic in situations where the data problem suffer from the lack the regularity. One possibility, how to overcome this obstacle, is to use the linearization employed in the nonlinear iterative solvers [2]. In this contribution, we present the framework of the goal-oriented error estimates based on this linearization. However, the linearization should be chosen carefully since the adjoint consistency has to be guaranteed. Several examples of differential equations with their discretizations are presented. This approach admits also to estimate the algebraic error arising from approximate solution of algebraic systems in a natural way and therefore we can easily control the iterative solvers. Moreover, we present the goal-oriented error estimates which take into account the geometry of mesh elements and which can be employed in the concept of the anisotropic hp-mesh adaptation, cf. [3]. The accuracy, efficiency and robustness of the presented approach is documented by several numerical experiments. We demonstrate by a numerical example that the adjoint problem based on a linearization of primal form provides qualitative similar error estimates as the original approach based on the differentiation.