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Basic Multigrid Solvers
Highest Rated
Rating: 4.6 out of 5(16 ratings)
175 students

Basic Multigrid Solvers

Computationally efficient solutions to sparse systems of linear equations.
Highest Rated
Created byRobert Spall
Last updated 1/2021
English

What you'll learn

  • Basics of multigrid solvers for large, sparse systems of linear equations.

Course content

3 sections • 15 lectures • 1h 0m total length

  • Introduction1:38

    A short introduction to the motivation for using multigrid solvers.

  • Download Notes and Codes0:31

    Which lectures contain downloadable materials.

  • Jacobi Iterative Technique5:51

    This lecture discusses the basics of the Jacobi iterative method for solving systems of equations.

  • Gauss-Seidel Iterative Technique3:58

    This lecture discusses the basics of the Gauss-Seidel iterative method for solving systems of equations.

  • Jacobi and Gauss-Seidel Error Reduction5:13

    This lecture discusses the high and low frequency error reduction properties of Jacobi, under-relaxed Jacobi, and Gauss-Seidel iterative methods.

  • Quiz1
  • Multigrid Basics5:06

    We look at an outline of a simple 2-level multigrid method.

  • Prototype One Dimensional Problem3:43

    We will define our prototype one-dimensional problem and look at our process for generating grids at different multigrid levels.

  • One Dimensional Results4:14

    We will look at rates of reduction of RMS residuals and error reductions for multigrid and Gauss-Seidel methods.

  • Options and Terminology3:16

    A look at different multigrid cycles, choices to be made, and terminology.

  • Different Multigrid Cycles
  • Different Multigrid Parameter Choices
  • Two Dimensional Results3:59

    Multigrid results for a 2D Poisson equation.

Requirements

  • Background in a scientific programming language and an understanding of basic numerical methods for solving linear systems.

Description

Multigrid techniques are used in most commercial computational fluid dynamics codes where large numbers of unknowns are common.  The techniques are used to accelerate convergence of basic iterative methods using multiple grid levels.  In this course we apply basic multigrid techniques to one- and two-dimensional elliptic problems discretized using a finite-difference method.  The approach may be extended to the finite-volume and other methods, or may be applied to general sparse linear systems of the form Ax=b.  The one- and two-dimensional codes are written in Fortran90 and source codes available for download.  Prospective students should be familiar with basic numerical methods and be proficient in a scientific programming language.

Who this course is for:

  • Users of commercial CFD solvers who would like some background on multigrid techniques used in the codes. Upper division undergraduates and beginning level graduate students in science and engineering.

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