30,283 research outputs found
Analysis of finite element approximation and iterative methods for time-dependent Maxwell problems
In this dissertation we are concerned with the analysis of the finite
element method for the time-dependent Maxwell interface problem when
Nedelec and Raviart-Thomas finite elements are employed and
preconditioning of the resulting linear system when implicit time schemes
are used.
We first investigate the finite element method proposed by Makridakis and
Monk in 1995. After studying the regularity of
the solution to time
dependent Maxwell's problem and providing approximation estimates for
the Fortin operator, we are able to give the optimal error estimate for the
semi-discrete scheme for Maxwell's equations.
Then we study preconditioners for linear systems arising in the finite
element method for time-dependent Maxwell's equations using implicit
time-stepping. Such linear systems are usually very large but sparse
and can only be solved iteratively. We consider overlapping Schwarz
methods and multigrid methods and extend some existing theoretical
convergence results. For overlapping Schwarz methods, we provide numerical
experiments to confirm the theoretical analysis
A Space-Time Discontinuous Galerkin Trefftz Method for time dependent Maxwell's equations
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
Dispersive properties of high order nedelec/edge element approximation of the time-harmonic Maxwell equations
The dispersive behaviour of high-order Næ#169;dæ#169;lec element approximation of the time harmonic Maxwell equations at a prescribed temporal frequency ω on tensor-product meshes of size h is analysed. A simple argument is presented, showing that the discrete dispersion relation may be expressed in terms of that for the approximation of the scalar Helmholtz equation in one dimension. An explicit form for the one-dimensional dispersion relation is given, valid for arbitrary order of approximation. Explicit expressions for the leading term in the error in the regimes where ωh is small, showing that the dispersion relation is accurate to order 2p for a pth-order method; and in the high-wavenumber limit where 1«ωh, showing that in this case the error reduces at a super-exponential rate once the order of approximation exceeds a certain threshold, which is given explicitly
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