Supporting calculation for the paper : Analysis of a Peaceman-Rachford ADI scheme for Maxwell equations in heterogenous media

The first eigenvalue of a one-dimensional transmission problem is calculated. The eigenvalue problem originates from the analysis of electromagnetic fields near an interior edge in a heterogeneous medium

A formula for the first positive eigenvalue of a one-dimensional transmission problem

A formula for the first positive eigenvalue of a one-dimensional elliptic transmission problem is derived. The eigenvalue problem arises during the spectral analysis of a Laplacian on the disc with angular transmission conditions. The formula for the first eigenvalue provides an explicit link to the transmission conditions in the problem

Interpolation of a regular subspace complementing the span of a radially singular function

We analyze the interpolation of the sum of a subspace, consisting of regular functions, with the span of a function with $r^\alpha$-type singularity. In particular, we determine all interpolation parameters, for which the interpolation space of the subspace of regular functions is still a closed subspace. The main tool is here a result by Ivanov and Kalton on interpolation of subspaces. To apply it, we study the $K$-functional of the $r^\alpha$-singular function. It turns out that the $K$-functional possesses upper and lower bounds that have a common decay rate at zero

A formula for the first positive eigenvalue of a one-dimensional transmission problem

A formula for the first positive eigenvalue of a one-dimensional elliptic transmission problem is derived. The eigenvalue problem arises during the spectral analysis of a Laplacian on the disc with angular transmission conditions. The formula for the first eigenvalue provides an explicit link to the transmission conditions in the problem

Analysis of a dimension splitting scheme for Maxwell equations with low regularity in heterogeneous media [revised]

We analyze a dimension splitting scheme for the time integration of linear Maxwell equations in a heterogeneous cuboid. The domain contains several homogeneous subcuboids, and serves as a model for a rectangular embedded waveguide. Due to discontinuities of the material parameters and irregular initial data, the solution of the Maxwell system has regularity below H$^{1}$. The splitting scheme is adapted to the arising singularities, and is shown to converge with order one in L$^{2}$. The error result only imposes assumptions on the model parameters and the initial data, but not on the unknown solution. To achieve this result, the regularity of the Maxwell system is analyzed in detail, giving rise to sharp explicit regularity statements. In particular, the regularity parameters are given in explicit terms of the largest jump of the material parameters. The analysis is based on semigroup theory, interpolation theory, and regularity analysis for elliptic transmission problems

ADI schemes for the time integration of Maxwell equations

This thesis is concerned with the analysis and construction of alternating direction implicit (ADI) splitting schemes for the time integration of linear isotropic Maxwell equations on cuboids. The work is organized in two major parts. The first part deals with time discrete approximations to exponentially stable Maxwell equations. By means of a divergence cleaning technique and artificial damping, we obtain an ADI scheme with approximations that also decay exponentially in time. The decay rate is here uniform with respect to the time discretization. One of the main ingredients in the proof of the decay behavior is an observability estimate for the numerical approximations. This inequality is obtained by means of a discrete multiplier technique. We also provide a rigorous error analysis for the uniformly exponentially stable ADI scheme, yielding convergence of order one in a space similar to $H^{-1}$. The error result makes only assumptions on the initial data and the model parameters. In the second part, we analyze time discrete approximations to linear isotropic Maxwell equations on a heterogeneous cuboid. In this setting, the domain consists of two different homogeneous subcuboids. The Maxwell equations are here integrated in time by means of the Peaceman-Rachford ADI splitting scheme which is well-known in literature. The main result provides a rigorous error bound of order 3/2 in $L^2$ for the numerical approximations. It is significant that the final error statement involves conditions only on the initial data and model parameters, but not on the solution. To achieve this result, we establish a detailed regularity analysis for the considered Maxwell system

Analysis of a dimension splitting scheme for Maxwell equations with low regularity in heterogeneous media

We analyze a dimension splitting scheme for the time integration of linear Maxwell equations in a heterogeneous cuboid. The domain contains several homogeneous subcuboids and serves as a model for a rectangular embedded waveguide. Due to discontinuities of the material parameters and irregular initial data, the solution of the Maxwell system has regularity below H$^1$. The splitting scheme is adapted to the arising singularities and is shown to converge with order one in L$^2$. The error result only imposes assumptions on the model parameters and the initial data, but not on the unknown solution. To achieve this result, the regularity of the Maxwell system is analyzed in detail, giving rise to sharp explicit regularity statements. In particular, the regularity parameters are given in explicit terms of the largest jump of the material parameters. The analysis is based on semigroup theory, interpolation theory, and regularity analysis for elliptic transmission problems

Analysis of a dimension splitting scheme for Maxwell equations with low regularity in heterogeneous media [revised]

We analyze a dimension splitting scheme for the time integration of linear Maxwell equations in a heterogeneous cuboid. The domain contains several homogeneous subcuboids, and serves as a model for a rectangular embedded waveguide. Due to discontinuities of the material parameters and irregular initial data, the solution of the Maxwell system has regularity below H$^{1}$. The splitting scheme is adapted to the arising singularities, and is shown to converge with order one in L$^{2}$. The error result only imposes assumptions on the model parameters and the initial data, but not on the unknown solution. To achieve this result, the regularity of the Maxwell system is analyzed in detail, giving rise to sharp explicit regularity statements. In particular, the regularity parameters are given in explicit terms of the largest jump of the material parameters. The analysis is based on semigroup theory, interpolation theory, and regularity analysis for elliptic transmission problems

Wellposedness and regularity for linear Maxwell equations with surface current

We study linear time-dependent Maxwell equations on a cuboid consisting of two homogeneous subcuboids. At the interface, we allow for nonzero surface charge density and surface current. This model is a first step towards a detailed mathematical analysis of the interactionof single-layer materials with electromagnetic fields. The main results of this paper provide several wellposedness and regularity statements for the solutions of the Maxwell system. To prove the statements, we employ extension arguments using interpolation theory, as well as semigroup theory and regularity theory for elliptic transmission problems

Analysis of a Peaceman–Rachford ADI scheme for Maxwell equations in heterogenous media

The Peaceman-Rachford alternating direction implicit (ADI) scheme for linear time-dependent Maxwell equations is analyzed on a heterogeneous cuboid. Due to discontinuities of the material parameters, the solution of the Maxwell equations is less than $H^2$-regular in space. For the ADI scheme, we prove a rigorous time-discrete error bound with a convergence rate that is half an order lower than the classical one. Our statement imposes only assumptions on the initial data and the material parameters, but not on the solution. To establish this result, we analyze the regularity of the Maxwell equations in detail in an appropriate functional analytical framework. The theoretical findings are complemented by a numerical experiment indicating that the proven convergence rate is indeed observable and optimal