3D-based hierarchical models and [formula omitted]-approximations for adaptive finite element method of Laplace problems as exemplified by linear dielectricity

Abstract

This paper is devoted to model adaptation and hpq-adaptive finite element methods for modeling and analysis of the problems for which the strong formulation corresponds to Laplace equation. The chosen example of this equation concerns dielectric structures (or media) of electrostatics. The paper addresses hierarchical theories (also called hierarchical models) and hierarchical approximations. In the assessment of the models and approximations, our own and existing a priori error estimation results are applied. The used assessment procedure can be employed to any other applications of Laplace equation in applied sciences. The proposed theories (understood as mathematical formulations) and their numerical approximations are applied to the physical model of linear dielectricity in structures with complex electric description and complex geometry. We take advantage of the 3D and 3D-based theories, hierarchical modeling, and hierarchical approximations within hpq finite element formulation. In our research, the applied theory and discretization parameters, i.e. the element size h, the longitudinal approximation order p, and the transverse order q, differ in each finite element.