A Generalized Discontinuous Galerkin (Gdg) Method and Its Applications

In paraxial approximations for wave propagations in optical waveguides, the time harmonic Maxwell's equations are approximated by Schrodinger equations where the propagation direction is identified as the time axis.

Due to this mismatch of refractive indices in waveguides, the electromagnetic fields are discontinuous solutions to Schrodinger equations, a property not shared by the probability wave functions of quantum mechanics.

In order to handle the discontinuities, we propose the idea of using generalized distribution variables in generalized discontinuous Galerkin (GDG) method.

The key idea is that, instead of using interior penalty terms in standard discontinuous Galerkin method, we propose to use delta functions as source terms to incorporate interface (jump) conditions into the given PDE.

The advantage of using the delta functions to enforce the interface conditions are three fold: (1) It can handle jump relationships of a general form. (2) The discontinuous Galerkin projection of the delta functions is natural due to the weak form definition of the distribution variables, with derived evenly split delta function and its related integration by parts formula. (3) The GDG approach can be easily extended to multi-dimensional problems and other types of PDEs of higher orders with nonsmooth solutions.

Numerical schemes for both 1-D and 2-D are formulated in details.

The consistence of the GDG scheme is proved for 1-D case. And we also showed the stability by calculating the eigenvalues of the discretization matrix.

Numerical results, with various types of jump conditions, demonstrate the GDG method's ability to handle general interface conditions and its high order accuracy.

As one of the GDG's application, second part of the dissertation proposes a new vectorial generalized discontinuous Galerkin beam propagation method (GDG-BPM) for wave propagations in inhomogeneous optical waveguides, where we combine the GDG method with the popular beam propagation method (BPM) by applying the GDG method on three coupled Schrodinger equations reduced from vector Helmholtz equation by paraxial approximation.

The resulting GDG-BPM takes on four formulations for either electric or magnetic field.

Numerical results, with different shapes of interface and amount of jumps, show the GDG-BPM's unique feature of handling interface jump conditions and its flexibility and high order accuracy in modeling wave propagations in inhomogeneous optical fibers.