401-0674-00L  Numerical Methods for Partial Differential Equations

 Semester Spring Semester 2020 Lecturers R. Hiptmair Periodicity yearly recurring course Language of instruction English Comment Not meant for BSc/MSc students of mathematics.

 Abstract Derivation, properties, and implementation of fundamental numerical methods for a few key partial differential equations: convection-diffusion, heat equation, wave equation, conservation laws. Implementation in C++ based on a finite element library. Objective Main skills to be acquired in this course:* Ability to implement fundamental numerical methods for the solution of partial differential equations efficiently.* Ability to modify and adapt numerical algorithms guided by awareness of their mathematical foundations.* Ability to select and assess numerical methods in light of the predictions of theory* Ability to identify features of a PDE (= partial differential equation) based model that are relevant for the selection and performance of a numerical algorithm.* Ability to understand research publications on theoretical and practical aspects of numerical methods for partial differential equations.* Skills in the efficient implementation of finite element methods on unstructured meshes.This course is neither a course on the mathematical foundations and numerical analysis of methods nor an course that merely teaches recipes and how to apply software packages. Content 1 Second-Order Scalar Elliptic Boundary Value Problems1.2 Equilibrium Models: Examples 1.3 Sobolev spaces1.4 Linear Variational Problems1.5 Equilibrium Models: Boundary Value Problems1.6 Diffusion Models (Stationary Heat Conduction)1.7 Boundary Conditions1.8 Second-Order Elliptic Variational Problems1.9 Essential and Natural Boundary Conditions2 Finite Element Methods (FEM)2.2 Principles of Galerkin Discretization2.3 Case Study: Linear FEM for Two-Point Boundary Value Problems2.4 Case Study: Triangular Linear FEM in Two Dimensions2.5 Building Blocks of General Finite Element Methods2.6 Lagrangian Finite Element Methods2.7 Implementation of Finite Element Methods2.7.1 Mesh Generation and Mesh File Format2.7.2 Mesh Information and Mesh Data Structures2.7.2.1 L EHR FEM++ Mesh: Container Layer2.7.2.2 L EHR FEM++ Mesh: Topology Layer2.7.2.3 L EHR FEM++ Mesh: Geometry Layer2.7.3 Vectors and Matrices2.7.4 Assembly Algorithms2.7.4.1 Assembly: Localization2.7.4.2 Assembly: Index Mappings2.7.4.3 Distribute Assembly Schemes2.7.4.4 Assembly: Linear Algebra Perspective2.7.5 Local Computations2.7.5.1 Analytic Formulas for Entries of Element Matrices2.7.5.2 Local Quadrature2.7.6 Treatment of Essential Boundary Conditions2.8 Parametric Finite Element Methods3 FEM: Convergence and Accuracy3.1 Abstract Galerkin Error Estimates3.2 Empirical (Asymptotic) Convergence of Lagrangian FEM3.3 A Priori (Asymptotic) Finite Element Error Estimates3.4 Elliptic Regularity Theory3.5 Variational Crimes3.6 FEM: Duality Techniques for Error Estimation3.7 Discrete Maximum Principle3.8 Validation and Debugging of Finite Element Codes4 Beyond FEM: Alternative Discretizations [dropped]5 Non-Linear Elliptic Boundary Value Problems [dropped]6 Second-Order Linear Evolution Problems6.1 Time-Dependent Boundary Value Problems6.2 Parabolic Initial-Boundary Value Problems6.3 Linear Wave Equations7 Convection-Diffusion Problems [dropped]8 Numerical Methods for Conservation Laws8.1 Conservation Laws: Examples8.2 Scalar Conservation Laws in 1D8.3 Conservative Finite Volume (FV) Discretization8.4 Timestepping for Finite-Volume Methods8.5 Higher-Order Conservative Finite-Volume Schemes Lecture notes The lecture will be taught in flipped classroom format:- Video tutorials for all thematic units will be published online.- Tablet notes accompanying the videos will be made available to the audience as PDF.- A comprehensive lecture document will cover all aspects of the course. Literature Chapters of the following books provide supplementary reading(detailed references in course material):* D. Braess: Finite Elemente,Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie, Springer 2007 (available online).* S. Brenner and R. Scott. Mathematical theory of finite element methods, Springer 2008 (available online).* A. Ern and J.-L. Guermond. Theory and Practice of Finite Elements, volume 159 of Applied Mathematical Sciences. Springer, New York, 2004.* Ch. Großmann and H.-G. Roos: Numerical Treatment of Partial Differential Equations, Springer 2007.* W. Hackbusch. Elliptic Differential Equations. Theory and Numerical Treatment, volume 18 of Springer Series in Computational Mathematics. Springer, Berlin, 1992.* P. Knabner and L. Angermann. Numerical Methods for Elliptic and Parabolic Partial Differential Equations, volume 44 of Texts in Applied Mathematics. Springer, Heidelberg, 2003.* S. Larsson and V. Thomée. Partial Differential Equations with Numerical Methods, volume 45 of Texts in Applied Mathematics. Springer, Heidelberg, 2003.* R. LeVeque. Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, UK, 2002.However, study of supplementary literature is not important for for following the course. Prerequisites / Notice Mastery of basic calculus and linear algebra is taken for granted.Familiarity with fundamental numerical methods (solution methods for linear systems of equations, interpolation, approximation, numerical quadrature, numerical integration of ODEs) is essential.Important: Coding skills and experience in C++ are essential.Homework assignments involve substantial coding, partly based on a C++ finite element library. The written examination will be computer based and will comprise coding tasks.