Anisotropic norm-oriented mesh adaptation for a Poisson problem

We present a novel formulation for the mesh adaptation of the approximation of a PDE. The discussion is restricted to a Poisson problem. The proposed formulation extends the goal-oriented formulation, since it is equation-based and uses an adjoint. At the same time, it supersedes it as a solution-convergent method. Indeed, goal-oriented methods 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 throughput 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. In the new proposed formulation, we amplify this advantage. We search, in the same well-identified set of metrics, the minimum of a norm of the approximation error. The norm is prescribed by the user and the method allows addressing the case of multi-objective adaptation, like, for example in aerodynamics, adapting the mesh for drag, lift, moment in one shot. In this work we consider the basic linear finite-element approximation and restrict our study to L norm in order to enjoy second-order convergence. Numerical examples for the Poisson problem are computed.