Introduction to Multiscale Mathematical Modeling

How can a model connect the fine structure of a material with the larger-scale behavior we can observe? Introduction to Multiscale Mathematical Modeling develops the mathematical tools for making that connection, while also examining where approximations work—and how their errors can be assessed.

Grigory Panasenko builds the discussion from core equations and analysis toward homogenization and models for thin, heterogeneous domains. Along the way, the book connects mathematical theory with applications in mechanics, materials science, and blood flow through vessel networks.

From mathematical foundations to multiscale methods

The opening chapters introduce equations of mathematical physics, boundary and initial conditions, and mathematical background including functional analysis. This groundwork supports the later analysis of partial differential equations, including questions of existence, uniqueness, and estimates.

Moving from the microscale to the macroscale

Homogenization is presented as a general approach to replacing fine-scale descriptions with effective larger-scale models. The exposition begins with a one-dimensional heat equation and extends to conductivity equations in multiple dimensions. Error estimates help clarify both the accuracy and the limits of these approximations.

Models that combine different dimensions

In its treatment of thin domains, the book explores asymptotic methods and partial domain decomposition. These approaches can combine a lower-dimensional model with three-dimensional regions where more detail is needed. The described applications include flow in vessel networks and modeling around a thrombus or stent.

For readers working across mathematics and applications

Based on courses for master’s and Ph.D. students, this book is suited to readers seeking a mathematically grounded introduction to multiscale methods. Its stated audience also includes engineers and readers in biophysics, alongside applied mathematicians interested in mechanics and heterogeneous materials.