# S5E2 – Non-standard Finite Element Methods

### Prof. Dr. Daniel Peterseim

#### Assistant: Dr. Mira Schedensack

Requirements: Basic knowledge in partial differential equations and finite element methods.

#### Description:

Non-standard finite element methods as non-conforming FEMs, mixed FEMs, or discontinuous Galerkin FEMs play an important role in practical applications as the Stokes equations from fluid dynamics, linear elasticiy from solid mechanics, or plate problems from structural mechanics. The seminar discusses various non-standard FEMs and their advantages in applications or implementation. Another focus lies on the error analysis. While the Galerkin orthogonality directly leads to a best-approximation result for conforming FEMs, new techniques are required to overcome the additional consistency error for non-standard FEMs. The recent medius analysis [3,1,2] shows equivalence of errors from non-conforming, mixed, discontinuous Galerkin, and conforming FEMs.

Further non-standard methods that can be discussed include finite volume methods, least-squares methods, boundary element methods, virtual element methods, or multi-scale FEMs.

The seminar will be based mainly on selected journal articles. Students interested in the seminar might contact M. Schedensack in advance to register.

#### Literature:

 [1] D. Braess. An a posteriori error estimate and a comparison theorem for the nonconforming $P_1$ element. Calcolo, 46(2):149–155, 2009. [2] 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. [3] T. Gudi. A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comp., 79(272):2169-2189, 2010.

 Date: Monday, 14(c.t.)–16, Wegelerstr. 6, SR 6.020 First seminar meeting: Monday, October 19th, 14(c.t.)–16, Wegelerstr. 6, SR 6.020.