IsoGeometric Mimetic Methods

Abstract

In this thesis I present a novel discretization procedure which combines two relatively new technologies for solving partial di_erential equations (PDE's): IsoGeometric Analysis (IGA) is a new paradigm which provides an exact geometry description and tight integration of Computer Aided Design (CAD) and Finite Element Analysis (FEA) by using the same basis for representation of the unknown _eld variables as is used for describing the geometry in CAD. Mimetic Discretization Methods on the other hand combine concepts from the Finite Element Method (FEM) and the Finite Volume Method (FVM) and provide a uni_ed and straightforward approach to model any physical _eld problem. Mimetic Methods aim at preserving as much as possible the structure of a PDE by 'mimicking' at the discrete level, important properties of the continuous realm, such that symmetries and conserved quantities are preserved. Central in this framework is the relation between physics and geometry. The Mimetic Discretization approach developed in this thesis is based upon B-splines1 for representing the unknown _eld variables. Besides inheriting all advantages from the IsoGeometric Analysis framework, B-splines appear as a natural basis for Mimetic Discretization Methods. They can be seen as higher order Whitney forms and provide vector spaces which are discretely conservative by construction. The resulting discretization approach resembles a Finite Volume Method on a staggered grid for the representation of the conservation laws and a Finite Element Method for the representation of the constitutive equations. In short, the scheme features the following advantages, - exact geometry description and tight integration with CAD; - fundamentally a higher order approach, featuring spectral like convergence. In practice though, IGA is con_ned to low or medium order due to bad conditioning of inner product mass matrices as a function of the polynomial order; - increased continuity resulting in continuous representations of _eld variables and derivatives; - a strong indication exists that these methods automatically meet the inf-sup conditions, leading to naturally stable discretizations of any physical problem; - local conservation of primal variables (strong) and secondary variables (weak); - in contrast to FVM and FEM which describe variables only locally, the Mimetic discretization is induced with a global topology which makes it possible to make useful decompositions of _eld variables.