# 401-3651-00L Numerical Analysis for Elliptic and Parabolic Partial Differential Equations

Semester | Autumn Semester 2020 |

Lecturers | C. Schwab |

Periodicity | yearly recurring course |

Language of instruction | English |

Comment | Course audience at ETH: 3rd year ETH BSc Mathematics and MSc Mathematics and MSc Applied Mathematics students. Other ETH-students are advised to attend the course "Numerical Methods for Partial Differential Equations" (401-0674-00L) in the CSE curriculum during the spring semester. |

Abstract | This course gives a comprehensive introduction into the numerical treatment of linear and nonlinear elliptic boundary value problems, related eigenvalue problems and linear, parabolic evolution problems. Emphasis is on theory and the foundations of numerical methods. Practical exercises include MATLAB implementations of finite element methods. |

Objective | Participants of the course should become familiar with * concepts underlying the discretization of elliptic and parabolic boundary value problems * analytical techniques for investigating the convergence of numerical methods for the approximate solution of boundary value problems * methods for the efficient solution of discrete boundary value problems * implementational aspects of the finite element method |

Content | The course will address the mathematical analysis of numerical solution methods for linear and nonlinear elliptic and parabolic partial differential equations. Functional analytic and algebraic (De Rham complex) tools will be provided. Primal, mixed and nonstandard (discontinuous Galerkin, Virtual, Trefftz) discretizations will be analyzed. Particular attention will be placed on developing mathematical foundations (Regularity, Approximation theory) for a-priori convergence rate analysis. A-posteriori error analysis and mathematical proofs of adaptivity and optimality will be covered. Implementations for model problems in MATLAB and python will illustrate the theory. A selection of the following topics will be covered: * Elliptic boundary value problems * Galerkin discretization of linear variational problems * The primal finite element method * Mixed finite element methods * Discontinuous Galerkin Methods * Boundary element methods * Spectral methods * Adaptive finite element schemes * Singularly perturbed problems * Sparse grids * Galerkin discretization of elliptic eigenproblems * Non-linear elliptic boundary value problems * Discretization of parabolic initial boundary value problems |

Literature | SUPPLEMENTARY Literature (core material will be in lecture notes) Brenner, Susanne C.; Scott, L. Ridgway The mathematical theory of finite element methods. Third edition. Texts in Applied Mathematics, 15. Springer, New York, 2008. xviii+397 pp. A. Ern and J.L. Guermond: Theory and Practice of Finite Element Methods, Springer Applied Mathematical Sciences Vol. 159, Springer, 1st Ed. 2004, 2nd Ed. 2015. R. Verfürth: A Posteriori Error Estimation Techniques for Finite Element Methods, Oxford University Press, 2013 Additional Literature: D. Braess: Finite Elements, THIRD Ed., Cambridge Univ. Press, (2007). (Also available in German.) Brezis, Haim Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. xiv+599 pp. D. A. Di Pietro and A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69 SMAI Mathématiques et Applications, Springer, 2012 [DOI: 10.1007/978-3-642-22980-0] V. Thomee: Galerkin Finite Element Methods for Parabolic Problems, SECOND Ed., Springer Verlag (2006). |

Prerequisites / Notice | Practical exercises based on MATLAB Former title of the course unit: Numerical Methods for Elliptic and Parabolic Partial Differential Equations |