Discontinuous and hybrid finite elements

News

• A preliminary discussion and the distribution of the seminar topics takes place on Thursday, Oktober 20th, 2016.

Date

Thu, 8:15 - 9:45, Arnimallee 7, SR 140 (Hinterhaus)

General Information

Description

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 [1]. 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)

Registration

Depending on your program of study, registration in the Campus Management is mandatory. In addition, all participants should register at the KVV.

Criteria for the Certificate

• active participation: a seminar talk incl. written summary (2-3 pages)
• constant participation

Talks

• 10.11.: The Nitsche-Method for the Laplace equation

• 24.11.: Saddle point problems, mixed finite elements and the Raviart-Thomas element for the Laplace equation, [2] III §4, §5 and [1] IV §1 1.2

• 1.12.: The interior penalty method for the Laplace equation, [3]

• 8.12.: Numerics to discontinuous Galerkin methods

Literature

[1] Franco Brezzi, Michel Fortin: Mixed and Hybrid Finite Element Methods. Springer (1991)

[2] Dietrich Braess: Finite Elemente. Springer (1996)

[3] 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).

[4] Béatrice Rivière: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. SIAM (2008)

[5] Leszek Demkowicz and Jay Gopalakrishnan: Analysis of the DPG method for the Poisson equation. SIAM J. Numer. Anal., 49(5), 1788–1809 (2011)

[6] Jay Gopalakrishnan: Five Lectures on DPG Methods. Preprint arXiv: 1306.0557 (2014)

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