Where is the field, and where is it going?

**Douglas N. Arnold:***
Differential Complexes in Numerical Analysis*

The de Rham complex and related differential complexes have recently
come to play an important role in the numerical analysis of partial
differential equations. To obtain stable discretizations of PDE, it is
often essential that appropriate aspects of the structure of the
equations be reflected in the discretization. In many cases it has
been found that the differential geometric structure captured by a
differential complex is key element, and a discrete version of the
complex must underlie a good numerical method. In this talk I will
explore this situation with a number of examples including PDE arising
from electromagnetics, elasticity, and general relativity.

**Ivo Babuska:***
Treatment of the Uncertainties in Computational Mechanics*

Computational mechanics is typically based on the deterministic
formulation and deterministic input data. Nevertheless in reality
smaller or larger uncertainties are always present. Essentially the
are two ways to treat these uncertainties:

- The worst scenario approach;
- The stochastic formulation.

**Jerry L. Bona:***
Computation of Singularities and other Phenomena
in Solutions of Nonlinear Wave Equations*

A discussion will be given of the use of computational methods in the
investigation of questions arising in the theory of partial
differential equations. The principal point in view is not the
application of the equations in science or engineering. Rather, the
lecture will focus on questions in the theory of nonlinear dispersive
wave equations that arise naturally, and for which analytical
techniques were insufficient for their resolution, at least initially.
Particularly emphasized will be the interaction between computation
and theory.

**James H. Bramble:***
Computational Scales of Sobolev Norms*

Often in the numerical treatment of partial differential equations or
pseudo differential equations, Sobolev spaces play a central role. In
the formulation, analysis and development of efficient methods for the
solution of the resulting approximate problems, it frequently turns
out to be of interest to be able to compute efficiently Sobolev norms
of fractional order and sometimes also of negative order. For example,
the spaces *H ^{1/2}* and

**Franco Brezzi:***
Stabilizing Subgrids
*

**Luis A. Caffarelli:***
Homogenization of Solutions to Fully Non Linear Equations in Random Media
*

**Craig C. Douglas:***
Modeling and Computation of Sea Surface Heights in Complex Domains*

A mathematical model and computational results are presented for
wind driven ocean modeling based on the spectral ocean element method. The
method is robust, accurate over a many year simulation, and scales extremely
well on a wide variety of parallel computers including traditional
supercomputers and clusters.

**Thomas Yizhao Hou:***
Multiscale Computation and Modeling of Flows
in Strongly Heterogeneous Porous Media*

Many problems of fundamental and practical importance contain multiple
scale solutions. Direct numerical simulations of these multiscale
problems are extremely difficult due to the range of length scales in
the underlying physical problems. recently, we have introduced a
multiscale finite element method for computing flow transport in
strongly heterogeneous porous media which contain many spatial
scales. The method is designed to capture the large scale behavior of
the solution without resolving all the small scale features. This is
accomplished by constructing the multiscale finite element base
functions that incorporate local microstructures of the differential
operator. By using a novel over-sampling technique, we can reconstruct
small scale velocity locally by using the multiscale bases. This
property is used to develop a robust scale-up model for flows through
heterogeneous porous media. To develop a coarse grid model for
multi-phase flow, we propose to combine grid adaptivity with
multiscale modeling. We also introduce a new class of numerical
methods to solve stochastic PDEs which can be used in conjunction with
the multiscale finite element method. We will demonstrate that our
numerical method can be used to compute accurately high order
statistical quantities more efficiently than the traditional
Monte-Carlo method.

**Pierre-Louis Lions:***
A Deterministic Particle Method for Diffusion Equations
*

**Mitchell B. Luskin:***
Mathematical and Computational Modeling
for a Solid-Solid Phase Transformation*

We present a mathematical model and computational results for the
solid-solid phase transformation of a thin film as the film is
cyclically heated and cooled. We propose and utilize a surface energy
that allows sharp interfaces with finite energy and a Monte Carlo
method to nucleate the phase transformation since the film would
otherwise remain in metastable local minima of the energy.

**Ricardo Nochetto:***
An Adaptive Uzawa FEM for the Stokes Equation: Convergence
Without the Inf-Sup Condition*

We introduce and study an adaptive finite element method for the
Stokes system based on an Uzawa outer iteration to update the pressure
and an elliptic adaptive inner iteration for velocity. We show linear
convergence in terms of the outer iteration counter for the pairs of
spaces consisting of continuous finite elements of degree *k* for
velocity whereas for pressure the elements can be either discontinuous
of degree *k* -1 or continuous of degree *k* -1 and *k*. The
popular Taylor-Hood family is the sole example of stable elements
included in the theory, which in turn relies on the stability of the
continuous problem and thus makes no use of the discrete inf-sup
condition. We discuss the realization and complexity of the elliptic
adaptive inner solver, and provide consistent computational evidence
that the resulting meshes are quasi-optimal. (This work is joint with
E. Baensch and P. Morin.)

**J. Tinsley Oden:***
Modeling Error Estimation and Model Adaptivity*

There are two major sources of error in computer simulations of
physical events: approximation error, due to the inherent inaccuracies
incurred in the discretization of mathematical models of the events,
and modeling error, due to natural imperfections of mathematical
abstractions of real physical events. This lecture develops a general
theory for estimating modeling error in models of nonlinear continuum
mechanics. Applications to heterogeneous media, viscous
incompressible flow, and nonlinear viscoelasticity are described and
associated adaptive modeling schemes are presented. The combination
of a posteriori error estimates of both approximations error and
modeling error are discussed. (With Serge Prudhomme.)

**Chi-Wang Shu:***
Finite Difference and Finite Volume WENO Schemes*

WENO schemes are finite difference or finite volume schemes
which are uniformly high order accurate and non-oscillatory
even with strong shocks. They are especially suitable for
hyperbolic or convection dominated problems. In this talk
we will describe the designing principles of such schemes,
their relationship to, and advantage and disadvantage against
other methods such as the discontinuous Galerkin methods.
We will also describe some of our recent work jointly with
various colleagues on the development and applications of WENO
schemes.

**Mary F. Wheeler:***
The Douglas Legacy for Modeling Flow and Transport
in Porous Media*

In this presentation, we discuss several algorithms for modeling flow
and transport in a permeable medium which were motivated by earlier
results of Professor Jim Douglas, Jr. They include extensions of
mixed finite element methods; namely, the expanded and mortar
upscaling mixed finite element methods for multiphase flow and the
characteristics mixed method for transport. In addition, we will also
consider a family of schemes referred to as discontinuous Galerkin
methods which are quite similar to Douglas' work on interior penalty
Galerkin methods. Both theoretical and computational results will be
presented. (With Todd Arbogast, Malgorzata Peszynska, and Beatrice Riviere.)

**Jinchao Xu:***
Asymptotically Exact a Posteriori Estimator and Superconvergence
for Unstructured Grids*

A new class of a posteriori error estimators will be presented for
general finite element solutions. Inspired by ideas from multigrid
methods, the new technique consists of taking average of the finite
element solution gradient followed by a few iteration of smoothings
that are often used in a multigrid process. This smoothed-averaged
gradient is shown to be superclose to the exact solution gradient.
As a result one obtains an a posteriori error estimator that can
proven to be asymptotically exact for very general unstructured grids.
A closely related result to be reported is a new superconvergence
estimate for linear finite element solution on most "practically
general" grids. This new superconvergence result is then used to
explain why some existing a posteriori error estimators work well in
some applications and why our new technique is useful in general.
(This work is joint with R. Bank.)

**Current and Future Trends in Numerical
PDE's****,**

*A conference in honor of the 75th Birthday of
Professor Jim Douglas, Jr.,*

*Last modified: 8 February 2002.*