Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Variational Formulation: Finite Element Method

In course

MATH-451: Numerical approximation of PDEs

The course is about the derivation, theoretical analysis and implementation of the finite element method for the numerical approximation of partial differential equations in one and two space dimensions. ...

Description

This lecture covers the variational formulation of the heat equation using the finite element method, discussing the discretization of the heat equation, variational formulation, and different numerical integration techniques such as Explicit Euler, Implicit Euler, and Crank-Nicholson method.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructor

Annalisa Buffa

Official source

Ontological neighbourhood

Mathematics

Analysis: Numerical analysis

Related concepts (25)

Euler method

In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book Institutionum calculi integralis (published 1768–1870).

Explicit and implicit methods

Explicit and implicit methods are approaches used in numerical analysis for obtaining numerical approximations to the solutions of time-dependent ordinary and partial differential equations, as is required in computer simulations of physical processes. Explicit methods calculate the state of a system at a later time from the state of the system at the current time, while implicit methods find a solution by solving an equation involving both the current state of the system and the later one.

Numerical methods for ordinary differential equations

Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation.

Numerical integration

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for numerical integration, especially as applied to one-dimensional integrals.

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.

Related lectures (52)

Explicit Stabilised Methods: Stochastic Differential Equations

Covers explicit stabilized methods for stiff stochastic differential equations, analyzing their properties and applications.

Ordinary Differential Equations: Methods and Applications

Explores ordinary differential equations and numerical integration methods for stability and accuracy.

Numerical Methods: Boundary Value Problems

Covers numerical methods for solving boundary value problems, including applications with the Fast Fourier transform (FFT) and de-noising data.

Numerical Methods for Boundary Value Problems

Covers numerical methods for solving boundary value problems using finite difference, FFT, and finite element methods.

Heat Equation: Modeling and Numerical Methods