In this presentation we give an overview of the research efforts in developing mixed finite element approximations of conservation laws. We develop hp adaptive H(div) conforming spaces in one, two and three dimensions by combining vector fields with H^1 conforming spaces. By the fact that the vector fields are generated using Piola transformations the H(div) conforming spaces are applicable to two dimensional manifolds and/or nonlinear geometric maps. It is shown that by increasing the internal order of approximation of elements with order k boundary fluxes, convergence rates of order h^k+1 for flux and order H^k+2 for pressure are obtained.
Arbitrary orders of approximation H^k+n for div(σ) can be obtained by further increasing the internal order of approximation.
H(div) approximations can, similarly to H^1 approximations, benefit from the use of quarterpoint element mappings for the resolution of singularities.
H(div) approximations with internal bubble functions naturally lead to a procedure for computing highly efficient error estimators.
We combine H(div) approximations with a multiscale hybrid mixed (MHM) approximation method to obtain a multiscale approximation method with local conservation.
All numerical results were obtained by algorithms implemented in the NeoPZ programming environment that is freely available from github http://github.com/labmec/NeoPZ
Speaker Affiliation: Distinguished Professor, Department of Mathematics, The University of Tennessee Distinguished Scientist & Group Leader, Computational & Applied Mathematics, Oak Ridge National Laboratory
This tutorial will focus on compressed sensing approaches to sparse polynomial approximation of complex functions in high dimensions. Of particular interest to the UQ community is the parameterized PDE setting, where the target function is smooth, characterized by a rapidly decaying orthonormal expansion, whose most important terms are captured by a lower (or downward closed) set. By exploiting this fact, we will present and analyze several procedures for exactly reconstructing a set of (jointly) sparse vectors, from incomplete measurements. These include novel weighted $\ell_1$ minimization, improved iterative hard thresholding, mixed convex relaxations, as well as nonconvex penalties. Theoretical recovery guarantees will also be presented based on improved bounds for the restricted isometry property, as well as unified null space properties that encompass all currently proposed nonconvex minimizations. Numerical examples are provided to support the theoretical results and demonstrate the computational efficiency of the described compressed sensing methods.
In the framework of steady-state diffusion problems, we show how the ideas of static condensation and hybridization lead to the introduction of the hybridizable discontinuous Galerkin methods.
Professor Cockburn received his Ph.D from University of Chicago in 1986 under the direction of Jim Douglas, Jr. He has spent all his academic career at University of Minnesota where he is now Distinguished McKnight University Professor. His research interests include the development of Discontinuous Galerkin methods for nonlinear conservation laws, second-order elliptic problems, electro-magnetism, wave propagation and elasticity.