################################################################# N U M E R I C A L A N A L Y S I S S E M I N A R Department of Mathematics, University of Maryland, College Park ################################################################# S P R I N G 1 9 9 4 S C H E D U L E ----------------------------------------------------------------------------- Jan. 27: A level set approach to computing solutions to incompressible two-fluid flows ----------------------------------------------------------------------------- Dr. Peter Smereka, UCLA A level set approach for computing solutions to incompressible two-fluid flow is presented. The interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function. We solve Navier-Stokes equation using a second order projection method which implements an ENO-type procedure for differencing the convection terms. A new treatment of the level set method allows us to include large density and viscosity ratios as well as surface tension. The method is Eulerian (using a fixed grid), easy to program, and allows topological changes with no added difficulty. We consider the motion of air bubbles in water and falling water drops in air. ----------------------------------------------------------------------------- Feb. 3: Adaptive pseudo-spectral methods with applications to shear band formation in viscoplastic materials ----------------------------------------------------------------------------- Dr. Dawn L. Crumpler, Northwestern University ----------------------------------------------------------------------------- Feb. 10: Constructing locking free elements for plates and shells ----------------------------------------------------------------------------- Prof. Douglas Arnold, Penn State University A variety of variational problems of mechanics involve energy functionals with multiple scales. In particular such problems often arise in models for thin structures such as beams, plates, and shells. Standard techniques of numerical approximation for such problems lead to extremely inaccurate results. This phenomenon is termed locking. Mixed variational formulations have proven to be the most successful means of avoiding locking, but they introduce difficulties of their own associated with the subtleties of discretizing saddle-point variational principles. Membrane locking in shell models has been particularly intractable. After reviewing the issues involved I will introduce a new method for deriving stable mixed finite element methods and describe its application to the approximation of plates and shell. ----------------------------------------------------------------------------- Feb. 17: High order finite element methods for elastoplastic problems ----------------------------------------------------------------------------- Li Yi-Wei, Institute for Physical Science and Technology, UMCP A high order h--version finite element method for two dimensional nonlinear elasto-plasticity problems will be presented. A family of admissible constitutive laws based on the so-called gauge function method is introduced first, and then a high order h-version semi-discretization scheme is presented. The existence and uniqueness of the solution for the semi-discrete problem are guaranteed by using some special properties of the constitutive law, and finally as the maximum element size h tends to zero, the solution of the semi-discrete problem will converge to the solution of the continuous problem. If the partition of the spatial space only has rectangles or parallelograms involved, then there would not be any limit on the element degree. However, if the partition of the spatial space has some triangular elements, then still certain combinations of high order finite element spaces for displacement and stress functions can be used. ----------------------------------------------------------------------------- Feb. 24: Algorithms for non-convex optimization in the calculus of variations ----------------------------------------------------------------------------- Dr. Noel J. Walkington, Carnegie Mellon University Numerical approximations of non-convex variational problems will be considered. The lack of convexity naturally leads one to consider highly oscillatory functions which are believed to model the fine scale microstructures ubiquitously observed in metallurgy. Since the oscillations occur on arbitrarily fine scales, they can not be directly modeled numerically. This motivates the development generalized solutions that characterize the fine scales using Young Measures, which, in principle, can be approximated on a macroscopic mesh. After some introductory examples, I will discuss various algorithms for the approximation of non-convex variational problems. In particular, I shall indicate what is known about the convergence of such algorithms and the practical difficulties that arise. ----------------------------------------------------------------------------- Mar. 3: On the efficient implementation of 3D boundary element methods ----------------------------------------------------------------------------- Dr. Stefan Sauter, Institute for Physical Science and Technology, UMCP Linear partial differential equations on a domain D can be transformed into boundary integral equations. At the first glance, the discretization of the boundary integral equations via the boundary element method (BEM) has some essential advantages compared with the direct discretization of the PDEs by FEM. The main advantages are: (1) There arises no problems if the domain D is unbounded. (2) The process of generating a finite element grid for BEM-equations is much more easy than for FEM discretizations, especially for complicated domains in 3D. The major drawbacks of the BEM are that the arising linear system contains a full coefficient matrix and that the generation of the corresponding matrix elements involves the evaluation of singular and nearly singular surface inte- grals. In our talk we will explain, how to avoid the generation and the storage of the full matrix by using the Panel-Clustering technique and how to evaluate the singular and nearly singular surface integrals very efficiently for the collocation and the Galerkin procedure. ----------------------------------------------------------------------------- Mar. 10: A projection method for reacting flow in the zero mach number limit ----------------------------------------------------------------------------- Dr. Mindy Lai, Lawrence Livermore National Laboratory A finite difference method is presented for solving the system of equations describing reacting flow in the zero Mach number limit. In this limit, acoustic waves are assumed to relax instantaneously. Moreover, these waves are weak and do not contribute significantly to the energetics of the flow. The appoach, therefore, is to solve a reduced system of equations where the detailed effects of the acoustics have been removed from the system. Despite this simplification, the reduced system allows for large temperatures and density variations, and correctly accounts for expansion due to heat release. The numerical method that has been developed is a modified projection method. It is second-order accurate in both space and time, and the time step is determined using only an advective CFL number. The method is a multi-step predictor-corrector formulation; in the first step, all nonlinear terms are calculated, and temperature, mean pressure, and density are estimated at the half time level. In the next step, the solution to the advection-reaction- diffusion equation is advanced in time to yield an intermediate velocity field which, in general, does not satisfy the divergence constraint. In the final step, this constraint is enforced, and scalar values are corrected. Calculations will be shown for the case of reacting flow in an enclosure. ----------------------------------------------------------------------------- Mar. 24: A Pointwise, A-Posteriori Error Estimator in the Finite Element Method ----------------------------------------------------------------------------- Dr. Jens Hugger, University of Copenhagen, Denmark When a boundary value problem possesses smooth solutions, it is possible very inexpensively to obtain a correction term to the finite element solution of a corresponding variational problem. The correction can be used for error estimation locally or globally in whatever norm is preferred, or in postprocessing of the solution to improve the quality. In this talk such a correction term is described for the general case of n-dimensional linear or nonlinear problems. Extensive computational evidence of the performance in one space dimension is given with special attention to the effects of the appearance of singularities and zeros of derivatives in the exact solution. A two-dimensional example shows the insufficiencies of the standard "bubble"-space approach. ----------------------------------------------------------------------------- Mar. 31: A Continuous Space-Time Finite Element Method for the Wave Equation ----------------------------------------------------------------------------- Dr. Todd Peterson, University of Virginia This talk will describe and analyze a discretization method for the wave equation which uses finite elements simultaneously in both space and time, thereby giving a unifed treatment of the spatial and temporal discretizations. The resulting space-time mesh is a tensor product mesh. This scheme has a discrete analogue of the energy conservation property of the exact equation. Connections with traditional finite element - finite difference schemes will be mentioned. By use of primarily variational arguments, existence, uniqueness, global error estimates and also higher order error estimates at time nodes will be established. Emphasis will be placed on the techniques used in the analysis. ----------------------------------------------------------------------------- Apr. 7: An Adaptive Projection Method for Low Speed Flows ----------------------------------------------------------------------------- Dr. Ann Almgren, Lawrence Livermore National Laboratory Projection methods using higher-order Godunov advection schemes have been used successfully to model incompressible flows. These methods can be extended to model low speed flows, such as those governed by the equations of the anelastic atmosphere. The solution of these equations in large and possibly complex domains requires not only efficiency in the basic algorithm but also adaptivity, to concentrate the resources where the highest resolution is needed. A representation of geometry is required to model terrain and/or obstacles in the flow. This talk will present a standard projection method using an approximate projection, the extension to an adaptive method using adaptive mesh refinement, and the inclusion of geometry using the Cartesian grid representation. The methods will be described in two dimensions, but extend naturally to three. ----------------------------------------------------------------------------- Apr. 14: Rounding Errors in Solving Block Hessenberg Systems ----------------------------------------------------------------------------- Dr. Urs von Matt, UMIACS, UMCP We analyse an algorithm proposed by G.W. Stewart for solving linear systems with block Hessenberg matrices. We will first review the algorithm which is based on a generalization of the Sherman-Morrison-Woodbury formula for the inverse of a modified matrix. Then we will consider its properties in the presence of rounding errors. Conditions are derived under which the algorithm computes a backward stable solution. ----------------------------------------------------------------------------- Apr. 21: On Some Techniques for the Approximation of Boundary Conditions in the Finite Element Method ----------------------------------------------------------------------------- Dr. Rolf Stenberg, Helsinki University of Technology, Finland We discuss the stabilization of finite element methods in which essential boundary conditions are approximated by Babuskas method of Lagrange multipliers. Several alternatives for the stabilization is considered. We show that there is a close connection with this technique and a classical method by Nitsche. ----------------------------------------------------------------------------- Apr. 28: Reordering Schemes for Sparse Matrix Decomposition ----------------------------------------------------------------------------- Aaron Naiman, Supercomputing Research Center, IDA Sparse matrices arise in many mathematical, physical and engineering problems, in techniques such as the finite element and finite difference methods. When the systems are solved directly by Gaussian elimination, different orderings of the matrix unknowns can lead to drastic differences in the required amounts of calculation and storage. In this talk we will survey yesterday and today's state-of-the-art reordering schemes for sparse matrix factorization. In addition, we will describe a code which compares the performance of different reordering schemes, together with some initial results. ----------------------------------------------------------------------------- May 5: Finite Element Methods for Nonlinear Parametrized PDEs ----------------------------------------------------------------------------- Felix Santos, Institute for Physical Science and Technology, UMCP NFEARS, a computer program aimed to the numerical analysis of parametrized nonlinear PDE's, has been in development during the past few years. I will describe the main features of this code, together with some of the underline theoretical aspects, which justifies them. Preliminary numerical results will be used in the discussion about the reliability of the results obtained by NFEARS, regarding in the most part with a-posteriori error estimation and adaptivity. Problems with singular points, such as turning points and bifurcation points, will be included in the numerical results.