Where is the field, and where is it going?

Titles and Abstracts of Talks

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:

  1. The worst scenario approach;
  2. The stochastic formulation.
The talk will address the problem of PDE's with stochastic input data and theoretical aspects of the formulation if the domain is described by a digital image.

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 H1/2 and H-1/2 naturally arise in boundary integral equations. Such spaces also appear naturally in some domain decomposition techniques for boundary value problems. The construction of bounded extension operators also may involve these spaces. In this talk I will describe how many such problems may be treated by means of multilevel techniques. Another application of this approach leads to uniform preconditioning of operators involving sums of operators of different orders.

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.