-FEM: From the Laplace Equation
Abstract:
The -FEM is a modern version of the Finite Element Method (FEM) capable of
extremely fast convergence through optimal combination of the size and polynomial
degree of elements. In this talk we present a couple of our recent results related
to elliptic problems, time-harmonic Maxwell's equations, incompressible flow, and
coupled problems. We begin with mentioning some new discrete maximum principles
for elliptic problems solved by the hp-FEM. Then we introduce a new class of
higher-order shape functions in the spaces 
and 
(curl) based on
generalized eigenfunctions of the Laplace and curl-curl operators. In both cases,
these shape functions possess a unique double orthogonality which makes them
close to optimal for the hp-FEM. Further we describe standard difficulties
with automatic hp-adaptivity and how they can be reduced using meshes with
arbitrary-level hanging nodes. Discussed will be the treatment of curvilinear
elements and why they lead to adaptive quadrature. At the end we describe our
first steps in the discretization of coupled problems on multiple meshes and
a new approach to space-time adaptivity via multi-mesh Rothe's method.
This seminar is easily accessible to persons with disabilities. For more information or for assistance, please contact the Mathematics Department at 743-3500.