Fem Fluids & Fun

Weighted Residuals, Functional Analysis, and Finite Elements for Incompressible Flow

From governing equations to weak formulations, function spaces, mixed finite elements, stabilization, and verification of incompressible-flow solvers.

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5thN Living Books · Working edition

Fem Fluids
& Fun

Weighted Residuals, Functional Analysis, and Finite Elements for Incompressible Flow

From governing equations to weak formulations, function spaces, mixed finite elements, stabilization, and verification of incompressible-flow solvers.

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Open manuscript · November 2026 edition

A living book. This manuscript is actively being written, expanded, and revised. Updates are generally published about once a month. Chapters and examples may change as the project develops.

What this book is trying to do

This book connects the physics of incompressible flow with the mathematical structure needed to approximate it. We start with conservation laws, derive weighted-residual and weak forms, interpret function spaces, and assemble the finite-element ideas used in Stokes and Navier–Stokes computations. The style emphasizes both the mathematics and why numerical methods succeed or fail.

Your reading path

Part IMathematical setting

Conservation laws, weighted residuals, and function spaces.

Part IIMixed finite elements

Weak forms, assembly, and stability.

Part IIIFlow computation

Time stepping, nonlinearity, verification, and next problems.

How to use the draft

Begin with the Start reading button, or choose a chapter in the left-hand table of contents. Each chapter offers working explanations, examples, and questions to guide future expansion. Use the search icon to locate a topic, and the next/previous arrows to move through the manuscript.

Scope of the starter R exercise: The included one-dimensional Poisson example teaches assembly mechanics; it is not a Navier–Stokes or incompressible-fluid-flow solver.

About the author and other books.