We consider the discretization of electromagnetic wave propagation problems by a discontinuous Galerkin method based on Trefftz polynomials. This method fits into an abstract framework for space-time discontinuous Galerkin methods for which we can prove consistency, stability, and energy dissipation without the need to completely specify the approximation spaces in detail. Any method of such a general form results in an implicit time stepping scheme with some basic stability properties. For the local approximation on every space-time element, we then consider Trefftz polynomials, i.e., the subspace of polynomials that satisfy Maxwell's equations exactly on the respective element. We present an explicit construction of a basis for the local Trefftz spaces in two and three dimensions and summarize some of their basic properties. Using local properties of the Trefftz polynomials, we can establish the well-posedness of the resulting discontinuous Galerkin Trefftz method. Consistency, stability, and energy dis...

A Space-Time Discontinuous Galerkin Trefftz Method for Time Dependent Maxwell's Equations

This thesis analyzes the discretization error induced by the Convolution Quadrature Galerkin method in seeking the numerical solution to time domain boundary integral equations arising in the problem of acoustic wave scattering by penetrable obstacles, electromagnetic wave scattering by a perfect electric conductor and heat conduction in the presence of a bounded inclusion. There are two sources of the numerical error: the error between the exact solution and spatial semidiscrete solution, and the error between the spatial semidiscrete solution and fully discrete solution. In the spatial semidiscrete error analysis, we fit the problem into the framework of the strongly continuous semigroup theory and obtain an error estimate sharper than that attainable by the traditional Laplace domain approach. For the full discrete error analysis, we try to apply the functional calculus theory to achieve a pure time domain approach with some success. Several numerical experiments are provided to validate the theoretical results.

Time domain boundary integral equation methods in acoustics, heat diffusion and electromagnetism

We consider the discretization of electromagnetic wave propagation problems by a discontinuous Galerkin Method based on Trefftz polynomials. This method fits into an abstract framework for space-time discontinuous Galerkin methods for which we can prove consistency, stability, and energy dissipation without the need to completely specify the approximation spaces in detail. Any method of such a general form results in an implicit time-stepping scheme with some basic stability properties. For the local approximation on each space-time element, we then consider Trefftz polynomials, i.e., the subspace of polynomials that satisfy Maxwell's equations exactly on the respective element. We present an explicit construction of a basis for the local Trefftz spaces in two and three dimensions and summarize some of their basic properties. Using local properties of the Trefftz polynomials, we can establish the well-posedness of the resulting discontinuous Galerkin Trefftz method. Consistency, stability, and energy dissipation then follow immediately from the results about the abstract framework. The method proposed in this paper therefore shares many of the advantages of more standard discontinuous Galerkin methods, while at the same time, it yields a substantial reduction in the number of degrees of freedom and the cost for assembling. These benefits and the spectral convergence of the scheme are demonstrated in numerical tests.

A Space-Time Discontinuous Galerkin Trefftz Method for time dependent Maxwell's equations

This paper investigates the numerical modeling of a time-dependent heat transmission problem by the convolution quadrature boundary element method. It introduces the latest theoretical development into the error analysis of the numerical scheme. Semigroup theory is applied to obtain stability in the spatially semidiscrete scheme. Functional calculus is employed to yield convergence in the fully discrete scheme. We compare the results to a more traditional Laplace domain approach.

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Time-domain boundary integral equation modeling of heat transmission problems

Time-domain electric integral equation methods (TD-EFIE) suffer from the numerical stability problem. Numerous schemes have been proposed over the years for the purpose of taking care of this instability. Among them, the marching-on-in-degree (MOD) method is one of the newly proposed algorithms for calculating the various transient responses. The MOD solver employs a set of associated Laguerre functions and incorporates exact temporal Galerkin testing. Although the MOD scheme is assumed to be more stable when compared with marching-on-in-time (MOT), there are few numerical or theoretical investigations on the stability of the various numerical recursive operations for the MOD methodology. It has been shown that the time-domain magnetic field integral equation (TD-MFIE) using the MOD method can be stabilized by using a Filon-type quadrature formula to accurately integrate the highly oscillatory Laguerre functions which has caused instability for the MOD solvers using a higher degree of approximation. On the other hand, the analytical expression of temporal derivatives used in the MOD scheme is strongly affected by an unbounded initial value of the transient current. In this paper, a novel Filon-type radial integration method and a sequential constraint term for TD-EFIE-MOD are proposed, which make the stabilized MOD solver accurate and stable for the arbitrary degree of the Laguerre polynomials. Some numerical results are presented to illustrate the validity of the stabilized MOD solution of TD-EFIE for transient scattering problems.

A Stabilized Marching-on-in-Degree Scheme for the Transient Solution of the Electric Field Integral Equation

Time domain integral equations have become a major tool in the computational analysis of electromagnetic scattering problems. Classically it was difficult to ensure a stable numerical solution of standard boundary integral formulations of the problem. In this chapter we shall discuss both theoretical and numerical aspects of one approach that solves the stability problem: convolution quadrature. We start with scattering from a perfectly conducting object and develop the electric field integral equation, as well as an error analysis of the fully discrete problem using finite elements in space. After presenting a brief discussion of some special numerical features of this problem, and some numerical results, we move on to scattering by a penetrable object. We end with a general discussion of computational electromagnetism illustrating the role that time domain integral equations can play.

Time Domain Integral Equation Methods in Computational Electromagnetism

Because the analysis of the time domain integral equations of electromagnetics is complicated and cannot easily take into account all sources of error in any given implementation, judgements of stability tend to rest on experience. While such an approach eliminates the worst methods immediately, subtle implementation issues may affect the stability of more promising approaches. In this work, we examine the effects of integration rules on stability. Numerical results demonstrate that increasing accuracy does not always increase stability.

Integration rules and experimental evidence for the stability of time domain integral equations

Because the mathematical analysis of time domain integral equations is fraught with difficulty, researchers are often forced to use experimental means to characterize the stability of their approaches. The difficulty inherent in this approach is that the variables affecting the success or failure of an experiment necessarily depend on a computer implementation of potentially hundreds of thousands of lines of instructions hiding unknown assumptions and even errors. In this work, the importance of different integration orders for near and far basis functions is investigated, and demonstrated to have an enormous effect on the stability of the implementation. Methods previously thought unstable for difficult problems are shown reliable with tiny changes over broad choices of parameter values.

Integration rules and time domain integral equation stability

In the analysis of time domain integral equations for the solution of electromagnetic problems, experience is the primary guide to choosing simulation parameters (such as integration rules and orders for the computation of kernel elements) that ensure stability. To improve this state of affairs, previous work examined the effects of integration orders on stability. In this work, adaptive quadrature will be used so that impact of integration accuracy on stability can be more finely assessed. Numerical results show that numerical integrals are not as reliable as practitioners often assume, and that high accuracy in the near field integration must be employed to achieve stability.

Integral accuracy and experimental evidence for the stability of time domain integral equations

The stability of time-domain (TD) integral equation (TDIE) approaches to the computation of electromagnetic scattering is profoundly affected by the accuracy of the underlying numerical integration methods used for computation of the kernel elements. In most publications, the accuracy of the quadratures used to compute kernel elements is never measured, and higher order computation is assumed to deliver high accuracy. In this communication, we examine the complicated relationship between the actual accuracy of kernel element computation and the resulting stability of integral equations. The numerical results show that stability may be improved for higher integral accuracy. Although integral accuracy is not always improved by higher order integration rules, more careful integration (as delivered by adaptive integration methods) is often helpful. The numerical results for a range of problems demonstrate these contentions.

Jielin Li

Papers

Time Domain Integral Equation Methods in Computational Electromagnetism

Integration rules and experimental evidence for the stability of time domain integral equations

Integration rules and time domain integral equation stability

Integral accuracy and experimental evidence for the stability of time domain integral equations

Integral Accuracy and the Stability of Two Methods for the Solution of Time-Domain Integral Equations for Scattering From Perfect Conductors