Energy-Norm Based and Goal-Oriented Automatic hp-Adaptivity. Application to High Frequency Electromagnetics

Energy-Norm Based and Goal-Oriented Automatic hp-Adaptivity. Application to High Frequency Electromagnetics

The Finite Element Method (FEM) enables the use of “adapted” meshes, not only to the geometry of the problem domain,but to the solution of the problem itself. Thus, very accurate solutions can be obtained, or equivalently, solutions with a given degree of accuracy using a minimum number of unknowns. Specifically, when the adaption of the mesh to the solution is made automatically, it is referred to as automatic adaptivity. The combination of h (change of the size of the elements) and p (change of polynomial order of approximation) refinements, i.e., hp-adaptivity provides exponential rates of convergence, even in the presence of singularities, in contrast to h and p schemes, in which algebraic rates of convergence are, in general, obtained.

The talk will present an automatic hp-adaptivity methodology based on the minimization of the energy norm and also based on the minimization of the error on a given quantity of interest (goal-oriented). Focus will be on applications for full vector Maxwell equations needed for the analysis of high frequency electromagnetic problems in the microwave and millimeter wave range. Specifically, results of the application to characterization of waveguiding problems, antenna radiation, and radar cross section, will be presented.