Multigrid Methods

Synopsis

Multigrid Methods: Axiomatic Convergence Theory for Linear and Weakly Nonlinear Problems (Multigrid Methods) is our new book (2026) on all things multigrid aimed at graduate and advanced undergraduate students and published by DeGruyter. Written by my colleague, Abner Salgado, Calvin Wong, and myself, Multigrid Methods is our attempt to make the subject accessible and understandable to a modern audience.

This book introduces the theory and application of multigrid methods for the fast numerical solution of linear and weakly nonlinear elliptic PDE. No previous exposure to numerical discretization methods is assumed. All that is required of the reader is curiosity and some basic knowledge of matrix theory and the theory of finite-dimensional vector spaces.

We use an axiomatic, mostly-matrix-based approach in the book, both as a way of presenting the theory in a natural and simple setting, and as a means for translating the theory into practical codes. We deviate a little from the matrix-based-approach in the presentation of the framework for nonlinear problems in the latter part of the book. That nonlinear analysis, based on subspace decompositions, represents an area of current research. In fact, the book takes the reader all the way from the basics and simple implementation issues to the front lines of multigrid research.

Coding the multigrid method is notoriously difficult. The current book, which contains several sample codes in the finite element and cell-centered finite difference frameworks, will train the interested reader in the construction of sophisticated, efficient multigrid codes using the simple but powerful MATLAB© programming environment.

Availability

The electronic version of the book can be found on De Gruyter’s website. Please ask your university library to purchase campus-wide access to the electronic version for you and your students. Physical copies can be purchased from Amazon.

Errata

You can get the errata file from the GitHub repo for the book here. Errors, typos and inconsistencies can be reported to either Abner or me. See my email address below. We greatly appreciate any feadback you have to give about the book.

Codes

There are several Matlab codes that are listed in the book, and these may be retrieved from the GitHub repo: stevenmwise/MultigridCourse. In addition, you can obtain the Matlab source codes used to generate the tables and figures dispayed in the book. We plan to add more software and supplementary material over time to the repo.