
Date Time 
Location  Speaker 
Title – click for abstract 

01/31 3:00pm 
BLOC 628 
Christian Klingenberg Würzburg University 
The inititial value problem for the multidimensional system of compressible gas dynamics may have infinitely many weak solutions
“We consider the isentropic compressible Euler equations in two space dimensions together with particular initial data. This data consists of two constant states only, where one state lies on the lower and the other state on the upper half plane. The aim is to investigate if there exists a unique entropy solution or if the convex integration method produces infinitely many entropy solutions. In this lecture we will show that the solution of this Riemann problem for the 2d isentropic Euler equations is nonunique (except if the solution is smooth). Next we are able to show that there exist Lipshitz data that may lead to infinitely many solutions even for the full system of Euler equations. This is joint work with Eduard Feireisl and Simon Markfelder. 

02/20 3:00pm 
BLOC 220 
Dmitri Kuzmin Dortmund University of Technology 
Boundspreserving limiters for continuous highorder finite element discretizations of hyperbolic conservation laws
In this talk, we constrain highorder finite element approximations to hyperbolic conservation laws using localized corrections to enforce discrete maximum principles. The use of Bernstein basis functions ensures that numerical solutions stay in the admissible range. The design of accuracypreserving FCT schemes for highorder Bernstein finite elements requires a major revision of algorithms designed for
loworder Lagrange elements. In this talk, we discretize the linear advection equation using an elementbased FCT algorithm which features: (i) a new discrete upwinding strategy leading to variation diminishing loworder approximations with compact stencils, (ii) a highorder stabilization operator based on the divergence of the difference between two gradient approximations, (iii) localized limiters for antidiffusive element contributions, and (iv) an accuracypreserving smoothness indicator that allows violations of strict maximum principles at smooth peaks. Additionally, we present limiters that constrain artificial diffusion coefficients or the difference between finite element basis functions
corresponding to highorder and piecewiselinear approximations. Extensions of FCT to hyperbolic systems will also be discussed. This is joint work with C. Lohmann, J.N. Shadid, S. Mabuza, and Manuel Quezada de Luna 

02/21 3:00pm 
BLOC 628 
John N. Shadid Sandia National Laboratories 
On Scalable Solution of Implicit FE Continuum Plasma Physics Models
The mathematical basis for the continuum modeling of plasma physics systems is the solution of the governing partial differential equations (PDEs) describing conservation of mass, momentum, and energy, along with various forms of approximations to Maxwell's equations. The resulting systems are characterized by strong nonlinear and nonsymmetric coupling of fluid and electromagnetic phenomena, as well as the significant range of time and lengthscales that these interactions produce. To enable accurate and stable approximation of these systems a range of spatial and temporal discretization methods are commonly employed. In the context of finite element spatial discretization methods these include mixed integration, stabilized and variational multiscale (VMS) methods, and structurepreserving (physics compatible) approaches. For effective longtimescale integration of these systems the implicit representation of at least a subset of the operators is required. Two wellstructured approaches, of recent interest, are fullyimplicit and implicitexplicit (IMEX) type timeintegration methods employing NewtonKrylov type nonlinear/linear iterative solvers. To enable robust, scalable and efficient solution of the largescale sparse linear systems generated by a Newton linearization, fullycoupled multilevel preconditioners are developed. The multilevel preconditioners are based on two differing approaches. The first technique employs a graphbased aggregation method applied to the nonzero block structure of the Jacobian matrix. The second approach utilizes approximate block factorization (ABF) methods and physicsbased preconditioning approaches that reduce the coupled systems into a set of simplified systems to which multilevel methods are applied. To demonstrate the flexibility of implicit/IMEX FE discretizations and the fullycoupled NewtonKrylovAMG solution approaches various forms of resistive magnetohydrodynamic (MHD) and multifluid electromagnetic plasma models are considered. In this context, we first briefl 

02/28 3:00pm 
BLOC 628 
Abner Salgado University of Tennessee 
Finite element approximation of nonconvex uniformly elliptic fully nonlinear equations
We propose and analyze a twoscale finite element method for the Isaacs equation. By showing the consistency of the approximation and that the method satisfies the discrete maximum principle we establish convergence to the viscosity solution. By properly choosing each of the scales, and using the recently derived discrete Alexandrov Bakelman Pucci estimate we can deduce rates of convergence. 

04/04 3:00pm 
BLOC 628 
Ridgway Scott Professor Emeritus, The University of Chicago 
Automated Modeling with FEniCS
The FEniCS Project develops both fundamental software components and enduser codes to automate numerical solution of partial differential equations (PDEs). FEniCS enables users to translate scientific models quickly into efficient finite element code and also offers powerful capabilities for more experienced programmers. FEniCS uses the variational formulation of PDEs as a language to define models. We will explain the variational formulations for simple problems and then show how they can be extended to simulate fluid flow. The variational formulation also provides a firm theoretical foundation for understanding PDEs. We argue that combining the theory with practical coding provides a way to teach
PDEs, their numerical solution, and associated modeling without requiring extensive mathematical prerequisites. As proof, this talk will require no background in PDEs or finite elements, only multivariate calculus. 

04/18 3:00pm 
BLOC 628 
Marta D’Elia Sandia National Laboratories 
An optimizationbased coupling strategy for local and nonlocal elasticity problems
Nonlocal continuum theories such as peridynamics and nonlocal elasticity can capture strong nonlocal effects due to longrange forces at the mesoscale or microscale. For problems where these effects cannot be neglected, nonlocal models are more accurate than classical Partial Differential Equations (PDEs) that only consider interactions due to contact. However, the improved accuracy of nonlocal models comes at the price of a computational cost that is significantly higher than that of PDEs. The goal of LocaltoNonlocal (LtN) coupling methods is to combine the computational efficiency of PDEs with the accuracy of nonlocal models. LtN couplings are imperative when the size of the computational domain or the extent of the nonlocal interactions are such that the nonlocal solution becomes prohibitively expensive to compute, yet the nonlocal model is required to accurately resolve small scale features. We propose an optimizationbased coupling strategy for the solution of a nonlocal elasticity problem. Our approach formulates the coupling as a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the nonlocal and local domains, and the controls are virtual volume constraints and boundary conditions. We present the implementation of our coupling strategy using Sandia's agile software components toolkit, which provides the groundwork for the development of engineering analysis tools. We show that our method passes linear and quadratic patch tests and we present numerical convergence studies. Using threedimensional geometries, we also show that our approach can be successfully applied to challenging, realistic, problems. 

04/25 3:00pm 
BLOC 628 
Jesse Chan Rice University 
Discretely entropy stable high order methods for nonlinear conservation laws
High order methods offer several advantages in the approximation of solutions of nonlinear conservation laws, such as improved accuracy and low numerical dispersion/dissipation. However, these methods also tend to suffer from instability in practice, requiring filtering, limiting, or artificial dissipation to prevent solution blow up. Provably stable finite difference methods based on summationbyparts (SBP) operators and a concept known as flux differencing address this inherent instability by ensuring that the solution satisfies a semidiscrete entropy inequality. In this talk, we discuss how to construct discretely entropy stable high order discontinuous Galerkin methods by generalizing entropy stable finite difference schemes using discrete L2 projection matrices and “decoupled” SBP operators. Extensions to curvilinear meshes will be also discussed, and numerical experiments for the one and twodimensional compressible Euler equations confirm the semidiscrete stability and high order accuracy of the resulting methods. 