Goal-oriented-error estimation for hierarchical models of a different type

Abstract

For the simulation of many physical processes, a collection of mathematical models with different level of sophistication is available. Such a collection is called a class of hierarchical models in which the hierarchy is determined by the level of sophistication of the models. In many such classes, solving a fine model is computationally costlier than solving a coarse model. Using the coarse model however, introduces a modelling error in the solution and hence in a quantity of interest. A new trend in computer simulations to reduce computational time but to meet a required accuracy is adaptive modelling. To drive an adaptive modelling process in which the accuracy of a quantity of interest is the goal, a goal-oriented modelling-error estimator is required. An approach to derive a goal-oriented modelling-error estimator is the Dual-Weighted Residual (DWR) method. The main question of this thesis is whether or not the DWR method is suitable for modelling error estimation in classes of hierarchical models in which the model equations are of a different type, i.e. when the models require different boundary conditions. For successful use of the DWR method, it is found to be essential to incorporate boundary residuals explicitly in the error estimator; especially for nonlinear problems where high-order boundary terms can be of significant magnitude.