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Abstract
Derivation, properties, and implementation of fundamental numerical methods for a few key partial differential equations, for instance, convection-diffusion, heat equation, wave equation, conservation laws, Stokes equations, Maxwell equations. Implementation in C++ based on a 2D 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
Second-Order Scalar Elliptic Boundary Value Problems Finite Element Methods (FEM) FEM: Convergence and Accuracy Second-Order Linear Evolution Problems Convection-Diffus Finite Elements for the Stokes Equation Finite-Element Exterior Calculus (FEEC)
Resources
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, seehttps://www.sam.math.ethz.ch/~grsam/NUMPDEFL/NUMPDE.pdf
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. * Douglas Arnold, Finite Element Exterior Calculus, SIAM, 2018. However, study of supplementary literature is not important for for following the course.