Mixed meshfree methods in computational electromagnetism: mathematical foundations and problems in wave scattering

Abstract

This thesis is primarily concerned with the extension of nodal meshfree methods to the solution of electromagnetic wave scattering problems in three dimensions. These problems involve vector field quantities, which are usually constrained by a divergence-free condition. The rather innocent addition of such a constraint on the divergence makes the analysis via nodal basis functions particularly challenging. In order to deal with it, we must add a Lagrange multiplier to the discretized weak forms. We are thus led to a mixed formulation which involves two quantities: The electric field and the Lagrange multiplier (also called pseudopressure). Next we investigate the conditions under which the aforementioned mixed formulation is well-posed; at this point the so-called inf-sup conditions play a fundamental role. After delving deeply on the theorems which comprise the framework of mixed formulations, one observes that the nodal approach we propose is backed by a firm mathematical theory. Finally, our meshfree formulation is put to the test by solving several problems pertaining to the subject of wave scattering