November 5 - 6, 2010
The most familiar family of finite elements is the Lagrange
family, which provide the canonical finite element approximation
of H1 on simplicial meshes in any dimension. In this talk we
discuss families of simplicial and cubical finite elements—some
previously known and some new—which are natural extensions
of the Lagrange family in various ways. Even for some of the
long known elements, a modern viewpoint based on the finite element
exterior calculus provides new properties and insights.
We consider a Cartesian PML approximation to solutions of
acoustic scattering
problems on an unbounded domain in ℝ^{2} and
ℝ^{3}. The perfectly
matched layer (PML)
technique modifies the equations outside of a bounded domain
containing the region
of interest. This is done in such a way that the new problem
(still on an unbounded
domain) has a solution which agrees with the solution of the
original problem. The
new problem has a solution which decays much faster, thus
suggesting replacing it
by a problem on a bounded domain. The perfectly matched layer
(PML) technique,
in a curvilinear coordinate system and in Cartesian
coordinates, has been studied for
acoustic scattering applications both in theory and
computation. Using a different
approach we extend the results of Kim and Pasciak concerning
the PML technique
in Cartesian coordinates. The exponential convergence of
approximate solutions as a function of domain size and/or the
PML "strength" parameter, σ_{0}
is also shown. We
note that once the stability and convergence of the
(continuous) truncated problem
has been established, the analysis of the resulting finite
element approximations is
then classical. Finally, the results of numerical computations
illustrating the theory,
in terms of efficiency and parameter dependence of the Cartesian
PML approach will
be given.
We study the lumped mass method for the model homogeneous heat
equation
with homogeneous Dirichlet boundary conditions. We first recall
that the maximum
principle for the heat equation does not carry over to the the
spatially semidiscrete
standard Galerkin finite element method, using continuous,
piecewise linear approximating functions. However, for the lumped mass variant the
situation is more
advantageous. We present necessary and sufficient conditions on
the triangulation, expressed in terms of properties of the stiffness matrix,
for the semidiscrete
lumped mass solution operator to be a positive operator or a
contraction in the
maximum-norm.
We then turn to error estimates in the L_{2}-norm. Improving
earlier results we
show that known optimal order smooth initial data error
estimates for the standard
Galerkin method carry over to the lumped mass method, whereas
nonsmooth initial
data estimates require special assumptions on the
triangulations.
We also discuss the application to time discretization by the
backward Euler and
Crank-Nicolson methods.