Multiscale finite element methods. Theory and applications.

*(English)*Zbl 1163.65080
Surveys and Tutorials in the Applied Mathematical Sciences 4. New York, NY: Springer (ISBN 978-0-387-09495-3/pbk; 978-0-387-09496-0/ebook). xii, 234 p. (2009).

Many scientific and engineering problems involve multiple scales. For instance, problems with composite materials or porous media lead to elliptic equations

\[ Lp := \div (k(x)\nabla p) = f, \]

and the coefficient \(k\) varies not only on a coarse scale. There is a fine structure due to the variation of \(k\) in each element of a triangulation for a finite element computation.

The main concept is similar to the upscaling/homogenization method. The basis functions of the finite element are adapted. While linear or bilinear finite elements satisfy \(\Delta\phi=0\) in the interior of the elements, here the relation \(L\phi=0\) is required (locally). Those functions are built into the Petrov-Galerkin method, finite volume method, and other discretization procedures. The aim is, of course, the approximative solution of the variational problem by computations on the coarse grid only.

Chapter 1 contains an introduction with challenging examples. Chapter 2 develops the method for linear problems with the intention that the reader feels at home when more involved problems will come. Chapter 3 is concerned with nonlinear equations. Behind the title of Chapter 4 ‘Multiscale finite element methods using limited global information’ there are hidden problems without a scale separation and therefore a simple adaption of local basis functions is not sufficient. Chapter 5 is devoted to applications to transport equations, Richards’ equation, fluid-structure interaction, oil reservoir modeling, and stochastic flows.

Each chapter starts with a description of the proposed method and their significance is demonstrated by numerical examples.

The convergence analysis of the multiscale method for a few representative cases is presented in Chapter 6, such that the theory is no burden for the reader who is mainly interested in applications. The analysis is based on a result by S. Moskow and M. Vogelius on homogenization [Proc. R. Soc. Edinb., Sect. A 127, No. 6, 1263–1299 (1997; Zbl 0888.35011)]. The discretization errors are given in terms of approximation errors with respect to standard Sobolev norms.

\[ Lp := \div (k(x)\nabla p) = f, \]

and the coefficient \(k\) varies not only on a coarse scale. There is a fine structure due to the variation of \(k\) in each element of a triangulation for a finite element computation.

The main concept is similar to the upscaling/homogenization method. The basis functions of the finite element are adapted. While linear or bilinear finite elements satisfy \(\Delta\phi=0\) in the interior of the elements, here the relation \(L\phi=0\) is required (locally). Those functions are built into the Petrov-Galerkin method, finite volume method, and other discretization procedures. The aim is, of course, the approximative solution of the variational problem by computations on the coarse grid only.

Chapter 1 contains an introduction with challenging examples. Chapter 2 develops the method for linear problems with the intention that the reader feels at home when more involved problems will come. Chapter 3 is concerned with nonlinear equations. Behind the title of Chapter 4 ‘Multiscale finite element methods using limited global information’ there are hidden problems without a scale separation and therefore a simple adaption of local basis functions is not sufficient. Chapter 5 is devoted to applications to transport equations, Richards’ equation, fluid-structure interaction, oil reservoir modeling, and stochastic flows.

Each chapter starts with a description of the proposed method and their significance is demonstrated by numerical examples.

The convergence analysis of the multiscale method for a few representative cases is presented in Chapter 6, such that the theory is no burden for the reader who is mainly interested in applications. The analysis is based on a result by S. Moskow and M. Vogelius on homogenization [Proc. R. Soc. Edinb., Sect. A 127, No. 6, 1263–1299 (1997; Zbl 0888.35011)]. The discretization errors are given in terms of approximation errors with respect to standard Sobolev norms.

Reviewer: Dietrich Braess (Bochum)

##### MSC:

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

35J25 | Boundary value problems for second-order elliptic equations |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |