Finite Elements I

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 peer-reviewed journals. He has supervised about 20 PhD students and 10 post-doctoral 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 I: Elements of Functional Analysis.- Lebesgue spaces.- Weak derivatives and Sobolev spaces.- Traces and Poincaré Inequalities.- Duality in Sobolev spaces.- Part II: Introduction to Finite Elements.- Main ideas and definitions.- One-dimensional finite elements and tensorization.- Simplicial finite elements.- Part III: Finite element interpolation.- Meshes.- Finite element generation.- Mesh orientation.- Local interpolation on affine meshes.- Local inverse and functional inequalities.- Local interpolation on non-affine meshes.- H(div) finite elements.- H(curl) finite elements.- Local interpolation in H(div) and H(curl) (I).- Local interpolation in H(div) and H(curl) (II).- Part IV: Finite element spaces.- From broken to conforming spaces.- Main properties of the conforming spaces.- Face gluing.- Construction of the connectivity classes.- Quasi-interpolation and best approximation.- Commuting quasi-interpolation.- Appendices.- Banach and Hillbert spaces.- Differential calculus.