(January 22) Dr. Andreas Veeser: Stability of Flat Interfaces During Semidiscrete Solidification - The stability of flat interfaces with respect to a spatial semidiscretization of a solidification model is analyzed. The considered model is the quasi-static approximation of the modified Stefan problem. The stability analysis relies on an argument developed by Mullins and Sekerka for the undiscretized case. The obtained stability properties differ from those with respect to the quasi-static model for certain parameter values and relatively coarse meshes. Moreover, consequences on discretization issues are discussed.

(January 29) Dr. Daniel Kessler : Numerical Analysis and Simulation of a Solutal Phase-Field Model - We characterize a binary alloy during isothermal solidification by its relative concentration and an order parameter for solidification: the phase-field. In an isotropic model, the evolution of these two quantities is described by a coupled parabolic system with Lipschitz continuous nonlinearities. Anisotropy depending on the direction of the gradient of the solution can also be added to the model. We discretize the system with P1 finite elements in space and a semi-implicit Euler scheme in time. The convergence of the scheme is proved in the isotropic case, using a generalized elliptic projector. Finally, for the anisotropic model we find a clear distinction between low and high anisotropy regimes via numerical simulations.

(February 5) Professor James Stone: Numerical Models of Accretion Flows Around Black Holes - Observations of black hole candidates at the center of the Milky Way and other galaxies indicate they radiate far less energy than expected based on the inferred rate of accretion of surrounding plasma. The resolution of this discrepancy may lie in the dynamics of the accreting plasma. It is now possible to compute the structure and evolution of accretion flows around compact objects such as black holes from first principles, by solving the equations of radiation magnetohydrodynamics using numerical methods. I will summarize the results of a series of numerical experiments which investigate the properties of time-dependent flows in 2D and 3D, and which are providing new insights into the physics of accretion flows.

(February 12) Professor Christoph Schwab: Two-scale Regularity and Sparse Grids for Homogenization Problems - We consider elliptic homogenization problems in periodic structures.

We present a two-scale regularity theory for the solutions of such homogenization problems in d-dimensional domains D (d=1,2,3) with coefficients and/or domains oscillating at period e < < diam(D).

We show that the solutions u(x,e) allow for high regularity independent of e if they are viewed as traces of certain functions in 2d-dimensions.

We show that tensorized Finite Elements in dimension 2d converge independently of e at the price of high complexity.

We show that a FEM based on sparse grids in 2d-dimensions converges independently of e in the physical domain D. The complexity of the sparse two-scale FEM is shown to equal, up to log(N), that of standard FEM in the physical d-dimensional domain on meshes of width e < h < 1.

The results are joint work with A.M. Matache and were obtained in the Project `Homogenization and Multiple Scales 2000'.

(February 19) Pingwen Zhang: Moving Mesh Methods Based on Harmonic Maps - In this talk, we will propose a moving mesh method which also contains two parts, a solution algorithm and a mesh-redistribution algorithm. A framework for adaptive meshes based on the Hamilton-Schoen-Yau theory was proposed by Dvinsky. We will extend Dvinsky's method to provide an efficient solver for the mesh-redistribution algorithm. The key idea is to construct the harmonic map between the physical space and a parameter space by an iteration procedure, and to update the interior and boundary grids simultaneously, rather than considering them separately. Application of the proposed moving mesh scheme is illustrated with some two- and three-dimensional problems with large gradients. The numerical experiments show that our methods can accurately resolve detail features of singular problems in 3D.

(February 26) Victoria E. Howle: Solving 3D Incompressible Navier-Stokes via Parallel Block Preconditioning - We consider parallel steady state and time dependent solutions of the 3D Lid Driven Cavity problem. This talk focuses mainly on two issues in resolving the set of nonlinear equations that result from discretization: (1) nonlinear solver performance and how different nonlinear solvers yield vastly different linear subproblem characteristics; (2) solution of the linear subproblems via block preconditioning along the lines of Kay & Loghin, Elman, and Silvester & Wathen. Specifically, both Newton's method and a fixed point technique based on an Oseen iteration are considered for the nonlinear problem. Block preconditioning methods using approximate Schur complement operators in conjuction with algebraic multigrid for subproblems will be discussed to solve the Oseen submatrix problems. For Newton's method, two preconditioners are considered: one applied to an explicitly calculated Jacobian matrix and the other applied to the Oseen matrix. We illustrate good convergence rates, similar to those observed by other authors, for two dimensional steady state and time dependent problems, time dependent three dimensional problems, and low Reynolds number steady state three dimensional problems. However, the steady state large Reynolds number 3D lid driven cavity problem is significantly more problematic. We will illustrate the utility of these ideas with a variety of benchmark problems and discuss some of the difficulties associated solution of steady 3D problems with large Reynolds numbers, as well as possible remedies.

(March 12) Prof. Zhimin Zhang: Recovery Techniques of the Finite Element Method - Recovery and post-processing are common practice in scientific and engineering computation with or without adaptivity. Some popular recovery methods, such as weighted averaging, local $L^2$-projection, the Zienkiewicz-Zhu patch recovery will be discussed and compared. Some superconvergence results will be presented.

(March 19) Richard S. Falk: Approximation by Quadrilateral Finite Elements - Approximation properties of quadrilateral finite element spaces of scalar functions and of vector-valued functions defined by the Piola transform are discussed. In particular, we consider neccessary conditions on the spaces of functions defined on the reference element for optimal order approximation on the physical element after bilinear mappings. As applications, we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for some standard finite element spaces.

(March 19) Prof. Alfred Schmidt: Multi-mesh finite element methods for coupled systems of PDEs with applications to thermo-elasticity and problems with phase transitions - Many models for physical problems lead to coupled systems of PDEs, where different components of the solution exhibit a strongly different local behavior in the underlying domain. An optimal discretization for a numerical simulation should be adapted to this behavior and use different locally refined meshes for different components. We present a general framework of adaptive finite element methods for coupled systems and show applications to problems from thermo-elasticity, semiconductor crystal growth with thermal convection, and phase field simulations for phase transitions.

(April 2) Prof. Tobias von Petersdorff: Boundary Element Methods for Maxwell Equations in Lipschitz Domains - We consider the scattering of electromagnetic waves at a bounded domain which can either be a perfect conductor or a dielectric material. We want to use a boundary integral formulation for the approximation, so that we only have to discretize the boundary of the scatterer. Until recently, only results for smooth domains were available (e.g. the book by Nedelec), but most applications have domains with edges and vertices.

We derive equivalent boundary integral formulations for both the perfect conductor and the dielectric case, and show that boundary element methods with Raviart-Thomas or Brezzi-Douglas-Marini elements converge quasioptimally.

The main difficulty for time-harmonic Maxwell equations is that electric and magnetic terms give opposite signs in the corresponding ``energy bilinear form''. In order to obtain a coercivity property one has to split functions into two parts using a ``Hodge decomposition''. For the discrete case we have to show that there is a discrete Hodge decomposition which is ``close'' to the continuous Hodge decomposition.

This is joint work with A. Buffa, R. Hiptmair, C. Schwab.

(April 9) Drs. Nail A. Gumerov and Ramani Duraiswami: Fast Multipole Methods for the Helmholtz Equation: New Fast Translation Operators - The 3-D Helmholtz equation arises in a number of important applications, including acoustical and electromagnetic wave scattering problems. The use of fast multipole techniques for the solution of this equation is an active and promising area of research. In this talk we will first present an overview of the FMM, with the goal of identifying the different parts of the algorithm, and the state-of-the-art in developing efficient implementations for these.

One of the most important issues in the FMM is the complexity of the translation operation. The practical implementation of an FMM algorithm and the cross-over point at which it begins to be useful, are critically dependent on this step. Previous authors have either accomplished translations with expensive algorithms, or with more inexpensive methods that use an approximate quadrature which causes additional errors and stability problems.

To solve this problem we derive new "translation theorems" for the fundamental solutions of the Helmholtz equation. Results of some multiple scattering problems using these theorems will be presented. Using the obtained relations we are able to derive a sparse matrix decomposition of the FMM translation operators for the Helmholtz equation. This decomposition allows evaluation of translations using the theoretical minimum of operations required. Our expressions do not require approximate quadrature as do those used in the literature, and are much more computationally efficient. We also can derive tighter error-bounds and complexity estimates for the Helmholtz FMM.

(Support of NSF Award 0086075 is gratefully acknowledged)

(April 16) Professor John Osborn: On Principles for the Selection of Shape Functions for the Generalized Finite Element Method - Effective shape functions for the Generalized Finite Element Method should reflect the available information on the solution. This information is fuzzy because the solution is, of course, unknown, and, typically, the only available information is the solution's inclusion in various function spaces. It is desirable to select shape functions that perform robustly over a family of relevant situations. Quantitative concepts of robustness are introduced and discussed. We show in particular that, in one dimension, polynomials are robust when the only available information consist in inclusions in the usual Sobolev spaces. If some additional information is available, if, {\it e.g.,} the approximated function is constrained by certain boundary conditions, then polynomials may perform poorly---relative to the optimal shape functions for approximating functions satisfying the boundary conditions---and some other family of shape functions should be used. This talk is based on joint work with I. Babu\v{s}ka and U. Banerjee.

(April 23) Dr. Cheng Wang: A Fourth Order Numerical Solver for the PlaneGeostrophic Equations Using Staggered Grid - The numerical method system of the planetary geostrophic equations (PGE), which has played a central role in large-scale ocean circulation theory, is taken into consideration. Such system can be viewed as asymptotic description of the primitive equations (PEs) of the atmospheric and oceanic flow as the Rossby number approaches to zero. There is only one prognostic equation in the system, which is for the temperature field. The horizontal velocity field is statically determined by the planetary geostrophic balance, and the vertical velocity field is determined by incompressibility condition, along with its vanishing boundary condition.

A fourth order finite difference method for the PGEs using MAC grid is analyzed in the talk. Different physical variables are evaluated on staggered numerical grid points. At each time step (stage), the temperature transport equations is explicitly updated by fourth order long-stencil approximation to the convection and diffusion terms. Subsequently, the temperature gradient can be obtained by finite difference operation. The horizontal velocity field is solved by a fourth order approximation to a first order ODE in vertical direction at each fixed horizontal point by using the combined information of the discrete value of the temperature gradient and the vanishing vertically averaged horizontal velocity field in the earlier derivation. The vertical velocity field is solved by a compact fourth order scheme for a second order ODE in vertical direction at each fixed horizontal point, with the vanishing Dirichlet boundary condition at the top and bottom.

The proposed method is shown both numerically and theoretically to preserve stability and fourth order convergence. Some numerical results, including an accuracy check and a physical example of a large-scale thermocline circulation, are presented.

(May 2) Prof. Randolph E. Bank: A New Paradigm for Parallel Adaptive Meshing Algorithms - We present a new approach to the use of parallel computers with adaptive finite element methods. This approach addresses the load balancing problem in a new way, requiring far less communication than current approaches. It also allows existing sequential adaptive PDE codes such as pltmg and mc to run in a parallel environment without a large investment in recoding. In this new approach, the load balancing problem is reduced to the numerical solution of a small elliptic problem on a single processor, using a sequential adaptive solver, without requiring any modifications to the sequential solver. The small elliptic problem is used to produce {\em a posteriori} error estimates to predict future element densities in the mesh, which are then used in a weighted bisection of the initial mesh. The bulk of the calculation then takes place independently on each processor, with no communication, using the same sequential adaptive solver. Each processor adapts its region of the mesh independently, and a nearly load-balanced mesh distribution is usually obtained as a result of the initial weighted bisection. Only the initial fan-out of the mesh decomposition to the processors requires communication. Two additional steps requiring boundary exchange communication are employed after the individual processors reach an adapted solution, namely the construction of a global conforming mesh from the independent subproblems, followed by a final smoothing phase using the subdomain solutions as an initial guess. We present a series of convincing numerical experiments which illustrate the effectiveness of this approach. This work is joint with Michael Holst.

(May 3) Prof. Randolph E. Bank: Multigrid: From Fourier to Gauss - Multigrid iterative methods are one of the most significant developments in the numerical solution of partial differential equations in the last twenty years. Multigrid methods have been analyzed from several diverse perspectives, from Fourier-like spectral decompositions to approximate Gaussian Elimination, with each perspective yielding new insight. In this lecture we will introduce the multigrid method, and survey several of the tools used in its analysis.

(May 7) Prof. Phillip Colella: Adaptive Mesh Refinement for Multiphysics Problems - In this talk, we will discuss a variety of algorithmic ideas that arise in extending the block-structured adaptive mesh refinement approach of Berger and Oliger to time-dependent problems in which a mixture of elliptic, parabolic and hyperbolic behaviors appear, such as in imcompressible fluid flows, low-Mach-number combustion, and problems with volume-of-fluid representations of fixed and free surfaces. Some of the issues to be addressed include the formulation of appropriate coarse-fine matching conditions; the influence of refinement in time on matching conditions for elliptic and parabolic problems, and on free-stream preservation; and error estimators at refinement boundaries. We will also discuss some of the software design issues that arise in supporting a diverse collection of algorithms, applications, and users.

(May 14) Prof. Kunibert Siebert: Fully Localized A Posteriori Error Estimators and Barrier Sets for Contact Problems - We discretize the elliptic obstacle problem, which constitutes a model contact problem, with finite elements. Enforcing the unilateral constraint solely at the nodes, we obtain a practical, but in general non-conforming method. We present a~posteriori error estimators for the maximum norm error. They are reliable, exhibit optimal order, and, except for the resolution of the obstacle, vanish within the discrete contact set. Moreover, we use these estimators to construct a~posteriori barrier sets for free boundaries under a natural stability condition on the continuous problem. We illustrate these results with simulations in 2d and 3d showing the impact of localization in mesh grading within the contact set along with quasi-optimal meshes.

This is joint work with R.,H. Nochetto (University of Maryland) and A. Veeser (Universita degli Studi di Milano).