

Lecture 1 Absract:
Lecture 1
Lecture 2 Lecture 3 Lecture 4 Lecture 5
Lecture 1
Lecture 2 Lecture 3 Lecture 4 Lecture 5
About the Speaker: Prof. Ovall is a computational mathematician who specializes in the efficient numerical treatment of singular solutions of partial differential equations, and is recognized as an expert on a posteriori error estimation and selfadaptive approximation in finite element methods. After receiving his PhD under Randolph Bank from the University of California in 2004, Jeff spent three years as a postdoctoral research associate at the Max Planck Institute for Mathematics in the Sciences in Leipzig (Germany) under Wolfgang Hackbusch, and another two years at the California Institute of Technology under Oscar Bruno, before receiving his first faculty position at the University of Kentucky. In 2013, he joined the faculty at Portland State University, where he holds an endowed Maseeh professorship. Lecture 1 Title: "Auxiliary subpace error estimation for Galerkin Finite Element Abstract: Efficient and reliable a posteriori error estimation is an essential component of highperformance finite element computations. Such estimates are used not only to determine whether or not a computed solution is sufficiently accurate, but also to guide a selfadaptive feedback loop when it is not. Because of the importance of such estimates, a variety of approaches have been developed in different contexts, with varying degrees of theoretical and empirical support. We will consider a class of estimators in which an approximate error function is (cheaply) computed in a suitablychosen auxiliary space, and argue that it provides an accurate estimate of the true error in norm. This auxiliary subspace approach is a natural extension of older hierarchical basis approaches, but more readily allows for nonstandard choices of auxiliary spaces and nonselfadjoint operators. The construction and analysis of the error estimator is carried out in such a way as to make a clear path for similar approaches with other differential operators and discretizations. Numerical experiments on a collection of benchmark elliptic problems in two and three dimensions demonstrate the efficiency and robustness of the approach. Lecture: 2 Date (Day): 26.11.2014 (Wednesday) Title: "Finite element error estimation for a Schr\"odinger operator with Abstract: In this talk we consider both boundary value problems (BVPs) and eigenvalue problems (EPs) for a Schr\"odinger operator with inversesquare potential, $\Delta +c^2 r^{2}$, where $r$ is the distance to the origin, which is assumed to be in $\overline\Omega$. Such potentials naturally arise in mathematical some models of molecular physics, quantum cosmology, and (linearized) combustion, for example. The inversesquare potential $c^2 r^{2}$ not only gives rise to new sources of singularities in the solution of BVPs and EPs when $c>0$ (the case $c=0$ is the familiar Laplacian), but also requires a different approach to both analysis and numerical analysis. In particular, the solution is no longer in the familiar Hilbert space $H^1$, but is more naturally set in a weighted Sobolev space. Using lowest order finite elements, we argue that an approximate error function computed in an auxiliary subspace provides highly accurate estimates of the error in $H^1$ and $L^2$, and the Hessian of the solution in $L^1$. A variety of numerical experiments support our theoretical results. We also offer a direct convergence and effectivity comparison between geometricallygraded meshes, which are based on a priori knowledge of possible singularities in the solution, and adaptively refined meshes driven by local error indicators associated with our a posteriori error estimate. Extensions to the associate eigenvalue problem will also be discussed. Lecture: 3 Date (Day): 27.11.2014 (Thursday) Title: "A framework for robust estimation of error in eigenvalue computations of Abstract: Eigenvalue problems for differential operators arise in a variety of important applications, and can pose nontrivial computational challenges beyond those typical for source problems. In terms of a posteriori error theory, some of these challenges are due to the existence of repeated or tightly cluster eigenvalues. For example, it may be difficult or impossible in practice to know if two computed eigenvalues which are very close to each other are approximating a single true eigenvalue of multiplicity two, or two distinct eigenvalues which just happen to be very close. The situation is further complicated for nonselfadjoint operators, for which we generally have complex eigenvalues (so no canonical ordering can be given), and even the notion of multiplicity must be considered more carefully. In this talk, we propose an approach for estimating error in eigenvalue and eigenvector computations which is robust with respect to multiple or tightly clustered eigenvalues. The key results make use of recent work concerning Kato's Square Root Conjecture for operators, and provide ideal (but not yet computable) error estimates in a general Galerkin setting. These ideal error estimates nonetheless bridge the gap between error estimation for eigenvalue problems and error estimation for source problems, allowing one to take advantage of existing techniques for source problems. We provide a realization of this framework in the $hp$finite element setting, and provide a variety of numerical experiments which corroborate our theoretical claims.
Lecture 1 "An Introduction to Mixed Finite Elements" Lecture 2 "The Immersed Boundary Element Method: Mathematical Formulation & Numerical Approximation" Lecture 3 "On the Discrete Compactness of hp Finite Elements & Application to Maxwell Equations" Lecture 4 Speaker:
