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

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