## DUNE – the distributed and unified numerics environment.(English)Zbl 1483.65004

Lecture Notes in Computational Science and Engineering 140. Cham: Springer (ISBN 978-3-030-59701-6/hbk; 978-3-030-59704-7/pbk; 978-3-030-59702-3/ebook). ix, 616 p. (2020).
This book is about the Distributed and Unified Numerics Environment for numerical computation using finite element and finite volume methods. A brief review on such methods is given, and how to implement those methods in DUNE based on the Dune module. For instace, the complete implementations of a finite element code are given for the Poisson equation, and a finite volume method for a transport equation.
The author discusses the overall structure of Dune, and the functions of modules are discussed. For examples, the grid interface for single-process situations and grids distributed across more than one process are discussed. The dune-localfunctions module is presented for collection of finite element bases on single elements. Numerical quadrature is discussed in the dune-geometry module. A module provides unified access to functions and discrete function spaces. The functions interface follows the approach in the C++ standard library and differentiable functions and functions are defined on finite element grids. The dune-functions module provides various standard bases.
In the appendix, the Dune build system is introduced and the complete example program source codes are listed for discussion in the book.
Reviewer’s remark: This book is very good for beginners for studying and learning finite element and finite volume methods in Dune.

### MSC:

 65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis 65-04 Software, source code, etc. for problems pertaining to numerical analysis 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs 65N08 Finite volume methods for boundary value problems involving PDEs

### Keywords:

finite element method; finite volume method
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