Mesh Adaptation for Computational Fluid Dynamics, Volume 2

6Norm-Oriented Formulations

The previous chapters describe two important approaches for anisotropic mesh adaptation for PDEs, namely the feature-based approach and the goal-oriented approach. This chapter presents a third approach, the norm-oriented approach. The norm-oriented formulation extends the goal-oriented formulation since it is equation-based and uses an adjoint. At the same time, the norm-oriented formulation somewhat supersedes the goal-oriented approach since it is basically a solution-convergent method. Indeed, goal-oriented methods solely rely on the reduction of the error in evaluating a chosen scalar output with the consequence that, as mesh size is increased (more degrees of freedom), only this output is proven to tend to its continuous analog while the solution field itself may not converge. A remarkable quality of goal-oriented metric-based adaptation is the mathematical formulation of the mesh adaptation problem under the form of the optimization, in the well-identified set of metrics, of a well-defined functional. With the norm-oriented approach, this latter advantage is amplified by searching in the same well-identified set of metrics, the minimum of a norm of the approximation error. This norm then converges to zero when mesh size is increased. The type of norm is prescribed by the user and the method also addresses the case of multi-objective adaptation like, for example, in aerodynamics, adaptating the mesh for drag, lift and moment in one shot. Numerical ...