# Legacy display Course

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#### Lecture in summer term 2014:

# S5E1 – Medius analysis of non-conforming Finite Element Methods

### Prof. Dr. Daniel Peterseim

This seminar aims to advertise non-conforming finite element methods for the numerical solution of partial differential equation in solid and fluid mechanics. In this context, non-conforming means that the finite element shape functions are not admissible solutions of the PDE. This approach has a stabilizing effect in regimes where conforming methods fail, e.g., thin plate bending or linear elasticity of nearly incompressible materials.

However, for a long time, there were strong doubts about the accuracy of non-conforming schemes. Dietrich Braess even remarked in his popular textbook [2]: ”This corresponds with the practical observation that non-conforming elements are much more sensitive to near singularities”.

A novel tool in numerical analysis called medius analysis [4] shows that this statement is not true. Medius analysis combines arguments from the a priori and the a posteriori error analysis and shows for the first time that non-conforming methods are by no means inferior to conforming schemes. The new analysis even allows a thorough comparison of discretization schemes that are conceptually very different [1, 3], e.g., discontinuous and continuous Galerkin, finite volume and least-squares methods.

- [1] D. Braess.
*Finite elements*. Cambridge University Press, Cambridge, third edition, 2007. - [2] D. Braess.
*An a posteriori error estimate and a comparison theorem for the nonconforming P1 element*. Calcolo, 46(2):149–155, 2009. - [3] C. Carstensen, D. Peterseim, and M. Schedensack.
*Comparison results of finite element methods for the Poisson model problem*. SIAM J. Numer. Anal., 50(6):2803–2823, 2012. - [4] T. Gudi.
*A new error analysis for discontinuous finite element methods for linear elliptic problems*. Math. Comp., 79(272):2169–2189, 2010.

Date & time: | Monday, | 14.15–15.45 Uhr, | Wegelerstr. 6 (HRZ), SR 5.002 |

Begin: | Monday, | 7.4.2013 |