30 research outputs found
Partially Penalized Immersed Finite Element Methods for Elliptic Interface Problems
This article presents new immersed finite element (IFE) methods for solving
the popular second order elliptic interface problems on structured Cartesian
meshes even if the involved interfaces have nontrivial geometries. These IFE
methods contain extra stabilization terms introduced only at interface edges
for penalizing the discontinuity in IFE functions. With the enhanced stability
due to the added penalty, not only these IFE methods can be proven to have the
optimal convergence rate in the H1-norm provided that the exact solution has
sufficient regularity, but also numerical results indicate that their
convergence rates in both the H1-norm and the L2-norm do not deteriorate when
the mesh becomes finer which is a shortcoming of the classic IFE methods in
some situations. Trace inequalities are established for both linear and
bilinear IFE functions that are not only critical for the error analysis of
these new IFE methods, but also are of a great potential to be useful in error
analysis for other IFE methods
Analysis of optical waveguides with arbitrary index profile using an immersed interface method
A numerical technique is described that can efficiently compute solutions in
interface problems. These are problems with data, such as the coefficients of
differential equations, discontinuous or even singular across one or more
interfaces. A prime example of these problems are optical waveguides and as
such the scheme is applied to Maxwell's equations as they are formulated to
describe light confinement in Bragg fibers. It is based on standard finite
differences appropriately modified to take into account all possible
discontinuities across the waveguide's interfaces due to the change of the
refractive index. Second and fourth order schemes are described with additional
adaptations to handle matrix eigenvalue problems, demanding geometries and
defects