| Date: | January 31, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Christian Klingenberg, Würzburg University |
| Title: | The inititial value problem for the multidimensional system of compressible gas dynamics may have infinitely many weak solutions |
| Abstract: | “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 2-d isentropic Euler equations is non-unique (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. |
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| Date: | February 20, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 220 |
| Speaker: | Dmitri Kuzmin, Dortmund University of Technology |
| Title: | Bounds-preserving limiters for continuous high-order finite element discretizations of hyperbolic conservation laws |
| Abstract: | In this talk, we constrain high-order 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 accuracy-preserving FCT schemes for high-order Bernstein finite elements requires a major revision of algorithms designed for
low-order Lagrange elements. In this talk, we discretize the linear advection equation using an element-based FCT algorithm which features: (i) a new discrete upwinding strategy leading to variation diminishing low-order approximations with compact stencils, (ii) a high-order stabilization operator based on the divergence of the difference between two gradient approximations, (iii) localized limiters for antidiffusive element contributions, and (iv) an accuracy-preserving 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 high-order and piecewise-linear 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 |
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| Date: | February 21, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | John N. Shadid, Sandia National Laboratories |
| Title: | On Scalable Solution of Implicit FE Continuum Plasma Physics Models |
| Abstract: | 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 length-scales 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 structure-preserving (physics compatible) approaches. For effective long-time-scale integration of these systems the implicit representation of at least a subset of the operators is required. Two well-structured approaches, of recent interest, are fully-implicit and implicit-explicit (IMEX) type time-integration methods employing Newton-Krylov type nonlinear/linear iterative solvers. To enable robust, scalable and efficient solution of the large-scale sparse linear systems generated by a Newton linearization, fully-coupled multilevel preconditioners are developed. The multilevel preconditioners are based on two differing approaches. The first technique employs a graph-based aggregation method applied to the nonzero block structure of the Jacobian matrix. The second approach utilizes approximate block factorization (ABF) methods and physics-based 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 fully-coupled Newton-Krylov-AMG solution approaches various forms of resistive magnetohydrodynamic (MHD) and multifluid electromagnetic plasma models are considered. In this context, we first briefl |
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| Date: | February 28, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Abner Salgado, University of Tennessee |
| Title: | Finite element approximation of nonconvex uniformly elliptic fully nonlinear equations |
| Abstract: | We propose and analyze a two-scale 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. |
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| Date: | April 4, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Ridgway Scott, Professor Emeritus, The University of Chicago |
| Title: | Automated Modeling with FEniCS |
| Abstract: | The FEniCS Project develops both fundamental software components and end-user 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 multi-variate calculus. |
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| Date: | April 18, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Marta D’Elia, Sandia National Laboratories |
| Title: | An optimization-based coupling strategy for local and nonlocal elasticity problems |
| Abstract: | Nonlocal continuum theories such as peridynamics and nonlocal elasticity can capture strong nonlocal effects due to long-range 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 Local-to-Nonlocal (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 optimization-based 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 three-dimensional geometries, we also show that our approach can be successfully applied to challenging, realistic, problems. |
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| Date: | April 25, 2018 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Jesse Chan, Rice University |
| Title: | Discretely entropy stable high order methods for nonlinear conservation laws |
| Abstract: | 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 summation-by-parts (SBP) operators and a concept known as flux differencing address this inherent instability by ensuring that the solution satisfies a semi-discrete 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 two-dimensional compressible Euler equations confirm the semi-discrete stability and high order accuracy of the resulting methods. |
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