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Understanding and implementing the finite element method. (English) Zbl 1105.65112

Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 0-89871-614-4/pbk). xvi, 363 p. (2006).
The treatment of stationary elliptic differential equations by the finite element method in this book consists of four parts. I: The basic framework. II: Data structures and Implementation. III: Solving the finite element equations. IV: Adaptive Methods.
The first part contains an introduction into the theory of conforming elements. After the classical formulation of the Poisson equation and the Lamé equation, the weak equations with the associated minimization problem and the Galerkin method are discussed. The existence of a solution is based on the Riesz representation theorem. It is shown to the reader without proofs, how convergence can be achieved for Lagrange elements with polynomials.
Part II is probably the author’s favorite part. It is the aim to support the reader when he/she wants a quick MATLAB implementation. To this end the data structure for the representation of the mesh, the connection with the computation of the numerical integrals and several MATLAB functions are explained.
This part contains also a hint that is concerned with convergence and accuracy of the finite element discretization. The reader is warned not to deal nearly incompressible material with piecewise polynomials of low degree, since locking phenomena can spoil the solution. The usual notation of volume locking is certainly better than the term mesh locking here, since the small parameter and the poor approximation of the kernel of the divergence operator is the reason, and it is not the shape of the mesh.
In Part III, after the Cholesky factorization, the conjugate gradient method is given as a powerful iterative method since it makes that the effective condition number is the square root of the given condition number. Classical iterations are plugged in as preconditioners and as smoothing operators for the multigrid algorithm. The smoothing property of the damped Jacobi iteration is verified for the Poisson equation on a regular grid and the Fourier modes. This proof requires several pages and is probably the hardest proof in the book.
The last part refers to adaptive algorithms with local mesh refinement. There are two error estimators, i.e., a heuristic estimator modeling the curvature of the solution and the well-known residual error estimator.
As already indicated, the book aims at readers who want a quick implementation, for instance via MATLAB. Difficult problems that require nonconforming elements or mixed methods are not covered. Those readers will perhaps appreciate that heuristic arguments are given instead of proofs. This may be only a handicap for understanding the nature of multigrid methods or the essence of condition numbers in the conjugate gradient algorithm.
Thus the book will have its readers.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J25 Boundary value problems for second-order elliptic equations
65F10 Iterative numerical methods for linear systems
65F35 Numerical computation of matrix norms, conditioning, scaling
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs

Software:

DistMesh; Matlab; LAPACK
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