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## Sobolev Spaces

We have qualiied surfaces by using the classical ��-function spaces (i.e., by requiring that the difeomorphisms �� are members of suh a space). Correspondingly, one could presume that we use these function spaces to qualify the solution of the integral equations we are faced with. his, however, is not practical since these function spaces are deined by the strong derivative;10 a irst step to broaden the domain in whih we searh for a solution is to introduce the weak derivative. he function spaces associated with the weak derivative are the Sobolev spaces. What makes the Sobolev spaces we encounter in the description of the scatering and radiation problems particularly suitable is the fact they are Hilbert spaces, that is, they come with an inner product. his inner product corresponds to the physical energy of the solution, and by searhing for a solution in a Sobolev space, we ask for a solution with inite energy [HK97], a result one would reasonably expect from a physical point of view. his section follows closely the treatment in [Ste10].

### Maxwell’s Equations and heir Solution

Since the focus is on time-harmonic problems, we assume and suppress the timedependency exp(−i��) in the following, where � is the time and � the angular frequency. How the electric current density � and harge distribution �e excite the electric ield � and the magnetic ield � and how these ields interact is described by Maxwell’s equations [Har01] curl � = i�� , (3.1) curl� = −i�� + � , (3.2) div� = �e , (3.3) div � = 0 , (3.4) where � is the electric and � the magnetic lux density. he current density � and the harge distribution �e are not independent since the harge must be conserved, that is, they must satisfy14 div � = i��.

#### Electrostatics: Laplace’s Equation and Integral Equation Formulations

In this thesis, we will frequently encounter integral operators known from the solution of Laplace’s equation. his equation is obtained by considering that for � = 0 the electric ield is conservative. his allows to express the electric ield as the gradient of a scalar potential, that is, � = − grad �e. Combining � = − grad �e together with (3.3) and (3.9) and assuming �e = 0, we yield Laplace’s equation Δ�e = 0 .

**Table of contents :**

**I. Prelude • **

**1. Introduction •**

a) Fragmentary Review of Numerical Tehniques •

b) Computational Complexity and Ill-Conditioned Integral Equations •

c) Review of Preconditioning Tehniques •

d) Scope and Outline of the hesis •

**2. Mathematical Preliminaries • **

a) Notation •

b) Surfaces and Mathematical Operators •

c) Sobolev Spaces •

�) Generalized Derivatives, Distributions, and Sobolev Spaces •

�) Sobolev Spaces on Surfaces •

� ) Vector Sobolev Spaces •

**3. Electromagnetic heory and Integral Equation Formulations • **

a) Maxwell’s Equations and heir Solution •

�) Continuity Conditions •

�) Equivalence Principle •

� ) Electromagnetic Potentials, Green’s Function, and Mixed Potential Formulas •

b) Scatering by or Radiation from a PEC Object •

�) Electric Field Integral Equation •

�) Magnetic Field Integral Equation •

� ) Combined Field Integral Equation •

c) Electrostatics: Laplace’s Equation and Integral Equation Formulations

**4. Discretization of Boundary Integral Operators and Equations • **

a) Petrov-Galerkin heory •

b) Basis Functions •

c) Discretization of the Field Integral Equations •

d) On the Ill-Conditioning of System Matrices •

�) uasi-Helmholz Decompositions •

�) uasi-Helmholz Projectors •

**II. Hierarhical Bases on Structured and Unstructured Meshes • **

**5. Primal and Dual Haar Bases on Unstructured Meshes for the EFIE • **

a) Bakground •

b) Construction of the Generalized Haar Basis •

c) New Hierarhical Basis •

�) Scalar Potential Operator •

�) Vector Potential Operator •

� ) Proposed Preconditioner for the EFIE Operator •

d) Numerical Results •

e) Conclusion •

**6. Hierarhical Bases on Multiply Connected Objects for the EFIE • **

a) Bakground •

b) New Formulation Without the Searh for Global Loops •

c) Implementational Issues •

d) Numerical Results •

e) Conclusion •

**7. On the Hierarhical Preconditioning of the CFIE • **

a) Spectral Analysis of the EFIE •

b) Spectral Analysis and Preconditioning of the CFIE •

c) Numerical Results •

d) Conclusion •

**III. Calderón Multiplicative Preconditioners • **

**8. On a Reinement-Free Calderón Multiplicative Preconditioner for the EFIE • **

a) Bakground •

b) New Formulation •

c) heoretical Apparatus •

�) Vector Potential Operator •

�) Scalar Potential Operator •

� ) Preconditioned Electric Field Integral Equation •

d) Numerical Results •

e) Conclusion •

**9. A Hermitian, Positive Deinite, and Well-Conditioned CFIE • **

a) New Formulation •

b) Numerical Results •

c) Conclusion •

**IV. Finale • **

**10. Concluding Scientiic Postscript • **

**A. he Discretized Laplace-Beltrami and the Hypersingular Operator • **

Nomenclature •

**Bibliography • **