#### Math. FEM

# Mathematical Theory of Finite Element Methods and Iterative Solvers

**Dr. P. Gatto**

A class for students of Mathematics, CES, and Simulation Science.

##### Dates

Monday, 10:15 - 11:45 in 1090 (Rogowski) | 328

Wednesday, 10:15 - 11:45 in 1090 (Rogowski) | 328

Office hours: Wednesday 12:00 - 13:00 in 1090 | 332a, or by appointment

The lectures will be held in English. Notes will be provided by the instructor, and are intended as the main course material. They include a number of exercises that are complimentary to the material covered in class. The final exam will be written and consist of exercises selected from the notes. This class is worth 6 ECTS. Please register with the L2P-Room, available at RWTH E-Learning.
See also the CAMPUS Information.

##### Contents

This is intended as a course at the intersection between functional analysis, finite element methods, and numerical linear algebra. Its objective is to show how all three areas are essential to modern computational techniques. The first part of the course reviews the theory of elliptic boundary value problems. Among other results, we will present the Lax-Milgram lemma, the Babuska-Necas theorem, and the Fredholm alternative. Emphasis will be put on the differences between coercive and non-coercive problems. The second part is devoted to continuous Galerkin discretizations. We discuss concepts such as stability, optimality, and convergence, and introduce the concept of asymptotic stability for non-coercive problems. We also present some classical error estimates. The last part of the course focuses on the efficient solution of the linear systems the arise from the discretization. We will focus on iterative solvers. Time permitting, we will cover the Conjugate Gradient method, the Minimum Residual method, and the Generalized Minimum Residual method. As far as applications are concerned, we shall focus on problems that arise from fluid dynamics: the Laplace equation, the Convection-Diffusion equation, and the Stokes equations.

##### Literature

Demkowicz, L., Oden, J.T., *Applied Functional Analysis *, CRC Press, 2010

Elman, H., Silvester, D., Wathen, A., * Finite Elements and Fast Iterative Solvers with Applications in Incompressible Fluid Dynamics *, Oxford University Press, 2005.

Saad, Y., *Iterative Methods for Sparse Linear Systems*.

Golub, G.H., Van Loan, C.F., *Matrix Computations*, The John Hopkins University Press, 1996.

Ciarlet, Ph.G., *The Finite Element Method for Elliptic Problems*.