Finite Elements III

Alexandre Ern is Senior Researcher at Ecole des Ponts and INRIA in Paris, and he is also Associate Professor of Numerical Analysis at Ecole Polytechnique, Paris. His research deals with the devising and analysis of finite element methods and a posteriori error estimates and adaptivity with applications to fluid and solid mechanics and porous media flows. Alexandre Ern has co-authored three books and over 150 papers in peerreviewed journals. He has supervised about 20 PhD students and 10 postdoctoral fellows, and he has ongoing collaborations with several industrial partners.Jean-Luc Guermond is Professor of Mathematics at Texas A&M University where he also holds an Exxon Mobile Chair in Computational Science. His current research interests are in numerical analysis, applied mathematics, and scientific computing. He has co-authored two books and over 170 research papers in peer-reviewed journals.

• Part XII: First-order PDEs.- Friedrichs' systems.- Residual-based stabilization.- Fluctuation-based stabilization (I).- Fluctuation-based stabilization (II).- Discontinuous Galerkin.- Advection-diffusion.- Stokes equations: Residual-based stabilization-. Stokes equations: other stabilizations.- Part XIII: Parabolic PDEs.- Bochner integration.- Weak integration and well-posedness.- Semi-discretization in space.- Implicit and explicit Euler schemes.- BDF2 and Crank-Nicolson schemes.- Discontinuous Galerkin in time.- Continuous Petrov-Galerkin in time.- Analysis using inf-sup stability.- Part XIV: Time-dependent Stokes equations.- Weak formulations and well-posedness.- Monolithic time discretization.- Projection methods.- Artificial compressibility.- Part XV: Time-dependent first-order linear PDEs.- Well-posedness and space semi-discretization.- Implicit time discretization.- Explicit time discretization.- Part XVI: Nonlinear hyperbolic PDEs.- Scalar conservation equations.- Hyperbolic systems.- First-order approximation.- Higher-order approximation.- Higher-order approximation and limiting.