Snorre H. Christiansen

A Multiplicative Calderon Preconditioner for the Electric Field Integral Equation

In this paper, a new technique for preconditioning electric field integral equations (EFIEs) by leveraging Calderon identities is presented. In contrast to all previous Calderon preconditioners, the proposed preconditioner is purely multiplicative in nature, applicable to open and closed structures, straightforward to implement, and easily interfaced with existing method of moments (MoM) code. Numerical results demonstrate that the MoM EFIE system obtained using the proposed preconditioning converges rapidly, independently of the discretization density.

A dual finite element complex on the barycentric refinement

Given a two dimensional oriented surface equipped with a simplicial mesh, the standard lowest order finite element spaces provide a complex X* centered on Raviart-Thomas divergence conforming vector fields. It can be seen as a realization of the simplicial cochain complex. We construct a new complex Y* of finite element spaces on the barycentric refinement of the mesh which can be seen as a realization of the simplicial chain complex on the original (unrefined) mesh, such that the L 2 duality is non-degenerate on Y i × X 2-i for each i ∈ {0,1,2}. In particular Y 1 is a space of curl-conforming vector fields which is L 2 dual to Raviart-Thomas div-conforming elements. When interpreted in terms of differential forms, these two complexes provide a finite-dimensional analogue of Hodge duality.

A Preconditioner for the Electric Field Integral Equation Based on Calderon Formulas

We describe a preconditioning technique for the Galerkin approximation of the electric field integral equation (EFIE), which arises in the scattering theory for harmonic electromagnetic waves. It is based on a discretization of the Calderon formulas and the Helmholtz decomposition. We prove several properties of the method, in particular that it produces a variational solution on a subspace of the Galerkin space for which we have an LBB inf-sup condition. When the Krylov spaces associated with the continuous operators are nondegenerate we prove that the discrete Krylov spaces converge as the mesh refinement goes to zero; when, moreover, the EFIE is nondegenerate on the continuous Krylov spaces, the discrete Krylov iterates converge towards the continuous ones. We also argue that one might expect the continuous Krylov iterates to exhibit superlinear convergence of the form

SMOOTHED PROJECTIONS IN FINITE ELEMENT EXTERIOR CALCULUS

n \mapsto C^n(n!)^{-\alpha}

The electric field integral equation on Lipschitz screens: definitions and numerical approximation

for some C > 0 and

Topics in structure-preserving discretization

\alpha>0

A CONSTRUCTION OF SPACES OF COMPATIBLE DIFFERENTIAL FORMS ON CELLULAR COMPLEXES

. Finally, we illustrate the theory with numerical experiments.

Discrete Fredholm properties and convergence estimates for the electric field integral equation

The development of smoothed projections, constructed by combining the canonical interpolation operators defined from the degrees of freedom with a smoothing operator, have proved to be an eective tool in finite element exterior calculus. The advan- tage of these operators is that they are L 2 bounded projections, and still they commute with the exterior derivative. In the present paper we generalize the construction of these smoothed projections, such that also non quasi-uniform meshes and essential boundary conditions are covered. The new tool introduced here is a space dependent smoothing operator which commutes with the exterior derivative.

Foundations of Finite Element Methods for Wave Equations of Maxwell Type

Summary. We use the integral equation approach to study electromagnetic scattering by perfectly conducting (non-orientable) Lipschitz screens. The well-posedness of the electric field integral equation is derived. The Galerkin method for this problem is analysed in a general setting and optimal error bounds are proved for conforming finite elements in natural norms.

Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension

We develop the theory of mixed finite elements in terms of special inverse systems of complexes of differential forms, defined over cellular complexes. Inclusion of cells corresponds to pullback of forms. The theory covers for instance composite piecewise polynomial finite elements of variable order over polyhedral grids. Under natural algebraic and metric conditions, interpolators and smoothers are constructed, which commute with the exterior derivative and whose product is uniformly stable in Lebesgue spaces. As a consequence we obtain not only eigenpair approximation for the Hodge-Laplacian in mixed form, but also variants of Sobolev injections and translation estimates adapted to variational discretizations.