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Speakers: **Rebecca Durst, **Wilkinson Fellow in Scientific Computing, Mathematics & Computer Science Division, Argonne National Laboratory

** Paul D. Hovland**, Senior Computer Scientish and Strategic Lead for Research Partnerships, Mathematics & Computer Science Division, Argonne National Laboratory

Title: There and Back Again: Discrete Challenges in Continuous Problems

Abstract: In many cases, solving a continuous problem involves solving one or more discrete problems along the way. In this two-part seminar, we first discuss a divergence-free, isoparametric method for the Stokes problem on smooth domains. This method builds on the Scott-Vogelius finite element pair with arbitrary polynomial degree greater than two, which is known to be a divergence-free method when implemented on affine elements. In order to preserve the divergence free property in the isoparametric framework, we use the Piola transform to modify the functions in the discrete velocity space. We also distribute the edge degrees of freedom at particular quadrature points. These choices ensure weak continuity across element edges that allow us to prove that the method converges with optimal order in the energy norm. We also prove that the discrete velocity error has optimal convergence in L^2.

In Part II, we discuss some combinatorial problems that arise in the efficient computation of sparse Jacobians and Hessians. Given the sparsity patterns of these matrices, one can use graph coloring to determine rows or columns of the Jacobian or Hessian that can be computed simultaneously, reducing the overall computational cost. We survey the various types of graph coloring problems that arise and briefly discuss heuristics for solving those problems. Unfortunately, the sparsity pattern is not always given and most methods for determining the sparsity pattern are slow. We therefore introduce a randomized method for determining an over-approximation of the sparsity pattern using cheap probes.

**Zoom:** **https://arizona.zoom.us/j/89529282577**

**Bios**

**Rebecca Durst** did her graduate studies in the Division of Applied Mathematics at Brown University. After receiving her Ph.D. in 2022, she was a postdoctoral associate at the University of Pittsburgh before joining Argonne in July 2024. Her research background is in numerical PDEs, specifically finite element methods and multiphysics problems. She is currently the Wilkinson Fellow in Scientific Computing in the Mathematics and Computer Science Division at Argonne National Laboratory.

**Paul D. Hovland** is a Senior Computer Scientist and Strategic Lead for Research Partnerships in the Mathematics and Computer Science Division at Argonne National Laboratory. His research revolves around technologies for facilitating software development for high-performance scientific computing, including research in automatic differentiation, automatic empirical performance tuning, computer-aided verification for parallel programs, and program synthesis.