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

 Semester Frühjahrssemester 2016 Dozierende R. Hiptmair Periodizität jährlich wiederkehrende Veranstaltung Lehrsprache Englisch Kommentar Not meant for BSc/MSc students of mathematics.

 Kurzbeschreibung 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 Python in one dimension and in C++ in 2D. Lernziel Main skills to be acquired in this course:* Ability to implement advanced 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. Inhalt 1 Case Study: A Two-point Boundary Value Problem1.1 Introduction1.2 A model problem1.3 Variational approach1.4 Simplified model1.5 Discretization1.5.1 Galerkin discretization1.5.2 Collocation [optional]1.5.3 Finite differences1.6 Convergence2 Second-order Scalar Elliptic Boundary Value Problems2.1 Equilibrium models2.1.1 Taut membrane2.1.2 Electrostatic fields2.1.3 Quadratic minimization problems2.2 Sobolev spaces2.3 Variational formulations2.4 Equilibrium models: Boundary value problems3 Finite Element Methods (FEM)3.1 Galerkin discretization3.2 Case study: Triangular linear FEM in two dimensions3.3 Building blocks of general FEM3.4 Lagrangian FEM3.4.1 Simplicial Lagrangian FEM3.4.2 Tensor-product Lagrangian FEM3.5 Implementation of FEM in C++3.5.1 Mesh file format (Gmsh)3.5.2 Mesh data structures (DUNE)3.5.3 Assembly3.5.4 Local computations and quadrature3.5.5 Incorporation of essential boundary conditions3.6 Parametric finite elements3.6.1 Affine equivalence3.6.2 Example: Quadrilaterial Lagrangian finite elements3.6.3 Transformation techniques3.6.4 Boundary approximation3.7 Linearization [optional]4 Finite Differences (FD) and Finite Volume Methods (FV) [optional]4.1 Finite differences4.2 Finite volume methods (FVM)5 Convergence and Accuracy5.1 Galerkin error estimates5.2 Empirical Convergence of FEM5.3 Finite element error estimates5.4 Elliptic regularity theory5.5 Variational crimes5.6 Duality techniques [optional]5.7 Discrete maximum principle [optional]6 2nd-Order Linear Evolution Problems6.1 Parabolic initial-boundary value problems6.1.1 Heat equation6.1.2 Spatial variational formulation6.1.3 Method of lines6.1.4 Timestepping6.1.5 Convergence6.2 Wave equations [optional]6.2.1 Vibrating membrane6.2.2 Wave propagation6.2.3 Method of lines6.2.4 Timestepping6.2.5 CFL-condition7 Convection-Diffusion Problems 7.1 Heat conduction in a fluid7.1.1 Modelling fluid flow7.1.2 Heat convection and diffusion7.1.3 Incompressible fluids7.1.4 Transient heat conduction7.2 Stationary convection-diffusion problems7.2.1 Singular perturbation7.2.2 Upwinding7.3 Transient convection-diffusion BVP7.3.1 Method of lines7.3.2 Transport equation7.3.3 Lagrangian split-step method7.3.4 Semi-Lagrangian method8 Numerical Methods for Conservation Laws8.1 Conservation laws: Examples8.2 Scalar conservation laws in 1D8.3 Conservative finite volume discretization8.3.1 Semi-discrete conservation form8.3.2 Discrete conservation property8.3.3 Numerical flux functions8.3.4 Montone schemes8.4 Timestepping8.4.1 Linear stability8.4.2 CFL-condition8.4.3 Convergence8.5 Higher order conservative schemes [optional]8.5.1 Slope limiting8.5.2 MUSCL scheme8.6. FV-schemes for systems of conservation laws [optional] Skript Lecture documents and classroom notes will be made available to the audience as PDF. Literatur 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 AppliedMathematical 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 ofSpringer Series in Computational Mathematics. Springer, Berlin, 1992.* P. Knabner and L. Angermann. Numerical Methods for Elliptic and Parabolic Partial DifferentialEquations, volume 44 of Texts in Applied Mathematics. Springer, Heidelberg, 2003.* S. Larsson and V. Thomée. Partial Differential Equations with Numerical Methods, volume 45 ofTexts 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. Voraussetzungen / Besonderes 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 in MATLAB and C++ are essential.Homework asssignments involve substantial coding, partly based on a C++ finite element library. The written examination will be computer based and will comprise coding tasks.