$$ \newcommand{\half}{\frac{1}{2}} \newcommand{\tp}{\thinspace .} \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\Aex}{{A_{\small\mbox{e}}}} \newcommand{\x}{\boldsymbol{x}} \renewcommand{\u}{\boldsymbol{u}} \renewcommand{\v}{\boldsymbol{v}} \newcommand{\dfc}{\alpha} % diffusion coefficient \newcommand{\If}{\mathcal{I}_s} % for FEM \newcommand{\Ifb}{{I_b}} % for FEM \newcommand{\sequencei}[1]{\left\{ {#1}_i \right\}_{i\in\If}} \newcommand{\sequencej}[1]{\left\{ {#1}_j \right\}_{j\in\If}} \newcommand{\basphi}{\varphi} \newcommand{\baspsi}{\psi} \newcommand{\xno}[1]{x_{#1}} \newcommand{\dx}{\, \mathrm{d}x} \newcommand{\ds}{\, \mathrm{d}s} $$

Time-dependent variational forms

Hans Petter Langtangen [1, 2]

[1] Center for Biomedical Computing, Simula Research Laboratory
[2] Department of Informatics, University of Oslo

Oct 30, 2015


PRELIMINARY VERSION

Table of contents

Discretization in time by a Forward Euler scheme
      Time discretization
      Space discretization
      Variational forms
      Simplified notation for the solution at recent time levels
      Deriving the linear systems
      Computational algorithm
      Example using sinusoidal basis functions
      Comparing P1 elements with the finite difference method
Discretization in time by a Backward Euler scheme
      Time discretization
      Variational forms
      Linear systems
Dirichlet boundary conditions
      Boundary function
      Finite element basis functions
      Modification of the linear system
      Example: Oscillating Dirichlet boundary condition
Analysis of the discrete equations
      Fourier components
      Forward Euler discretization
      Backward Euler discretization
      Comparing amplification factors
Exercises
      Exercise 1: Analyze a Crank-Nicolson scheme for the diffusion equation
Bibliography