 INTERDYSCYPLINARY CENTRE FOR MATHEMATICAL AND COMPUTATIONAL MODELLING # Discretization of electromagnetic problems

Readership: Code-developers and users of numerical methods of 'finite element' and/or 'finite volume' flavor.

Level: Late undergraduate, graduate. Background in linear algebra and some familiarity with variational methods is assumed. No advanced notions of functional analysis (such as Sobolev spaces, etc.) are required.

What's in the course:

The main idea, which goes much beyond electromagnetics, is this: Many partial differential equations are just the local expression of some set of integral laws, that expresses the conservation of something. Hence, physical laws can be formulated as equalities between various kinds of integrals, such as line integrals, surface integrals, etc., and their time derivatives.

Now, if one is content with enforcing these conservation laws, not on all lines, surfaces, etc., but only on those generated by a discretization mesh, what one gets is a numerical scheme, naturally endowed with the desirable conservation properties. Something is lost, of course, in the process: Constitutive laws, as a rule, will not be satisfied exactly: this is where the discretization error lies, and the central problem with approximation methods.

A somewhat surprising but, as will be argued, logical consequence of holding this viewpoint, is that finite elements are not, primarily, "interpolants", that generate fields from degrees of freedom. We rather see them, here, as a device to produce approximations of lines, surfaces, etc., by mesh-related polygons, polyhedra, etc. Hence linear operators, the nature of which will be made precise in this course, whose adjoints, known as "Whitney forms", generalize standard finite elements in a useful way. How useful is demonstrated by example applications such as electrostatics and magnetostatics, whose mathematical structure can be found, identical up to mere changes of symbols, in many other branches of applied science.

Here follows the Table of Contents. Click on the appropriate button to download a .ps or .pdf version for each of the four Chapters and the reference list. The .ps files don't exceed 100O kO each. The .pdf files don't exceed 300 kO each. (Both versions are formatted alike, and run over 67 A4 pages.)

### 1. Geometric preliminaries       .ps.pdf

• Vector space, vector subspaces
• Group actions, homogeneous spaces.
• Affine space, barycentric coordinates
• Cells, piecewise smooth manifolds
• Orientation of space
• Oriented manifolds, inner and outer orientation
• Chains, boundary operator
• Metric notions, Euclidean structures

### 2. Rewriting the Maxwell equations    .ps.pdf

• Integration: Circulation, flux, etc.
• Differential forms, and their physical relevance
• The Stokes theorem
• Electric field and magnetic field, as a 1-form and a  2-form
• The Hodge operator
• The Maxwell equations: Discussion
• Boundary conditions, transmission conditions
• Wedge product, energy
• Underlying structures: the 'Maxwell house'

### 3. Discretizing       .ps.pdf

• A model problem: antenna radiating in closed cavity
• Primal mesh
• Dual mesh
• A discretization kit
• Network equations, discrete Hodge operator
• The toolkit
• Playing with the kit: full Maxwell
• Playing with the kit: statics
• Playing with the kit: miscellanies

### 4. Finite elements       .ps.pdf

• Consistency
• Stability
• The time-dependent case
• Whitney forms
• As a device to approximate manifolds
• A generating formula for
• Properties of
• Of higher degree
• For other shapes than simplices