WiSe 16/17: Seminar: Discontinuous and hybrid finite elements
Ralf Kornhuber, Maren-Wanda Wolf
Information for students
The preliminary discussion and distribution of seminar topics takes place on 20.10.2016, 08:00-10:00.close
Content: Mixed and hybrid finite element methods overcome locking phenomena in computational mechanics and allow for higher order approximations of dual variables like stress or flux rather than primal variables like displacement and pressure . Existence and convergence analysis relies on basic properties of constrained minimization and saddle point problems [1, 2].
More recently, discontinuous Galerkin (DG), discontinuous Petrov-Galerkin (DPG) methods, or corresponding hybrid versions HDG and HPDG have been developed and analyzed, exploiting the same mathematical structures [3, 4, 5, 6]. Combining local mass conservation with arbitrary order these methods have become the method of choice, e.g., in computational porous media flow.
In this seminar, we plan to highlight the basic ideas, pros and cons of these advanced discretization methods for partial differential equations and the mathematical background of their analysis.
Target audience: Students in the Master Course Mathematics or BMS (Phase I)
Prerequisites: Basic knowledge on theory and numerics of elliptic pdes as taught, e.g. in the lecture "Numerik von partiellen Differentialgleichungen (Numerik III)" at FU Berlinclose
 Franco Brezzi, Michel Fortin: Mixed and Hybrid Finite Element Methods. Springer (1991)
 Dietrich Braess: Finite Elemente. Springer (1996)
 Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, and L. Donatella Marini: Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems. SIAM J. Numer. Anal., 39(5), 1749–1779 (2002).
 Béatrice Rivière: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. SIAM (2008)
 Leszek Demkowicz and Jay Gopalakrishnan: Analysis of the DPG method for the Poisson equation. SIAM J. Numer. Anal., 49(5), 1788–1809 (2011)
 Jay Gopalakrishnan: Five Lectures on DPG Methods. Preprint arXiv: 1306.0557 (2014)close
16 Class schedule