A POD goal-oriented error measure for mesh optimization

The approach for designing an error measure to guide an adaptive meshing algorithm proposed in Power et al. (Ocean Modell. 2006; 15:3–38) is extended to use a POD adjoint-based method, thus facilitating efficient primal and adjoint integration in time. The aim is to obtain a new mesh that can adequately resolve all the fields at all time levels, with optimal (w.r.t. the functional) efficiency. The goal-based method solves both the primal and adjoint equations to form the overall error norms, in the form of a metric tensor. The tetrahedral elements are then optimized so that they have unit size in Riemannian space defined with respect to the metric tensor. This is the first attempt to use POD to estimate an anisotropic error measure. The metric tensor field can be used to direct anisotropic mesh adaptivity. The resulting mesh is optimized to efficiently represent the solution fields over a given time period. The calculation of the error measures is carried out in the reduced space. The POD approach facilitates efficient integration backwards in time and yields the sensitivity analysis necessary for the goal-based error estimates. The accuracy of both the primal and adjoint-reduced models is thus optimized (through the use of anisotropic mesh adaptivity). In addition, the functional for optimizing meshes has been designed to be consistent with that for 4D Var data assimilation. Copyright q 2009 John Wiley & Sons, Ltd.