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.