This tutorial is an introduction to the numerical solution of elliptic partial differential equations (PDEs) and parallelization techniques, with an emphasis on finite element methods (FEM). Thus, it covers an important part of computational science, since most of the models behind computer simulations are based on PDEs. The material is presented in eight chapters, which are, to a large extent, self-contained.
After a short introduction in chapter 1, the Poisson equation is used in chapter 2 to present first ideas about discretization methods for PDEs, basic solvers, and parallelization by data decomposition. An introduction to the basic terms and concepts of parallelism, such as memory consumption, communication requirements, and efficiency measures are given in chapter 3. There is a short summary of parallel programming with message passing operations, but without an example of a parallel program or a concrete description of the message passing interface (MPI) standard.
Chapter 4 is devoted to the solution of elliptic PDEs by finite element methods. The mathematical background of FEM is given, and includes fundamental theorems with proofs. The content of the chapter is exemplified by a heat conduction problem. Chapter 5 briefly introduces basic concepts for sparse storage schemes, data distribution with respect to data decomposition, and vector operations. No sequential or parallel programs are given, but the exercises guide the reader to develop MPI implementations for the algorithms introduced. Chapter 6 presents classical solvers for systems of linear equations generated by discretizing an elliptic PDE. Direct solvers, like LU and incomplete LU (ILU), and iterative solvers, like Jacobi and Gauss-Seidel, red-black Gauss-Seidel, alternate-direction implicit (ADI), and the conjugate gradient (CG) method (including generalized minimal residual (GMRES) and bi-conjugate gradient stabilized (BICGSTAB)), are covered, and data parallel implementations are sketched. Finally, chapter 7 summarizes the multigrid method, and provides hints for a parallel implementation. Chapter 8 lists further topics in the area of parallel solvers for PDEs, and a list of references with 116 entries provides many suggestions for further reading and additional material concerning parallelism and applied mathematics. An appendix includes more than 30 Web addresses for communication libraries, software tools in parallelism, libraries for numerical algorithms, or benchmarks. A list of abbreviations and an index are provided.
Chapters 3 through 7 each end with an exercise paragraph containing a guideline for practical programming exercises, which altogether results in an entire parallel implementation of an FEM solver, based on the multigrid method. The book itself contains no parallel programming examples, but a practical course can be downloaded from the Web address provided.
The book is intended for advanced undergraduate and graduate students in computational science and engineering. Given the presentational style of the tutorial, knowledge in applied mathematics is needed if the book is to be used for self-study. For a classroom tutorial, additional material on parallel programming with MPI and further mathematical background material might be necessary.
In summary, this tutorial provides a concise introduction to basic concepts in solving PDEs using parallel methods, with an emphasis on the mathematical derivation of those methods. Due to the limited length of the book, the description of parallelism and parallel algorithms is brief, and might require supplementary material. The tutorial can be a starting point for work in the area of parallel PDE solvers.
