The spring semester 2021 will generally take place online. New presence elements as of April 26 will be communicated by the lecturers.

# 263-5001-00L  Introduction to Finite Elements and Sparse Linear System Solving

 Semester Autumn Semester 2017 Lecturers P. Arbenz Periodicity yearly recurring course Language of instruction English

 Abstract The finite element (FE) method is the method of choice for (approximately) solving partial differential equations on complicated domains. In the first third of the lecture, we give an introduction to the method. The rest of the lecture will be devoted to methods for solving the large sparse linear systems of equation that a typical for the FE method. We will consider direct and iterative methods. Objective Students will know the most important direct and iterative solvers for sparse linear systems. They will be able to determine which solver to choose in particular situations. Content I. THE FINITE ELEMENT METHOD (1) Introduction, model problems. (2) 1D problems. Piecewise polynomials in 1D. (3) 2D problems. Triangulations. Piecewise polynomials in 2D. (4) Variational formulations. Galerkin finite element method. (5) Implementation aspects.II. DIRECT SOLUTION METHODS (6) LU and Cholesky decomposition. (7) Sparse matrices. (8) Fill-reducing orderings.III. ITERATIVE SOLUTION METHODS (9) Stationary iterative methods, preconditioning. (10) Preconditioned conjugate gradient method (PCG). (11) Incomplete factorization preconditioning. (12) Multigrid preconditioning. (13) Nonsymmetric problems (GMRES, BiCGstab). (14) Indefinite problems (SYMMLQ, MINRES). Literature [1] M. G. Larson, F. Bengzon: The Finite Element Method: Theory, Implementation, and Applications. Springer, Heidelberg, 2013.[2] H. Elman, D. Sylvester, A. Wathen: Finite elements and fast iterative solvers. OUP, Oxford, 2005.[3] Y. Saad: Iterative methods for sparse linear systems (2nd ed.). SIAM, Philadelphia, 2003.[4] T. Davis: Direct Methods for Sparse Linear Systems. SIAM, Philadelphia, 2006.[5] H.R. Schwarz: Die Methode der finiten Elemente (3rd ed.). Teubner, Stuttgart, 1991. Prerequisites / Notice Prerequisites: Linear Algebra, Analysis, Computational Science.The exercises are made with Matlab.