In this book the author sets out to answer two important questions: 1. Which numerical methods may be combined together? 2. How can different numerical methods be matched together? In doing so the author presents a number of useful combinations, for instance, the combination of various FEMs, the combinations of FEM-FDM, REM-FEM, RGM-FDM, etc. The combined methods have many advantages over single methods: high accuracy of solutions, less CPU time, less computer storage, easy coupling with singularities as well as the complicated boundary conditions. Since coupling techniques are essential to combinations, various matching strategies among different methods are carefully discussed. The author provides the matching rules so that optimal convergence, even superconvergence, and optimal stability can be achieved, and also warns of the matching pitfalls to avoid. Audience: The book is intended for both mathematicians and engineers and may be used as text for advanced students.

In this book the author sets out to answer two important questions: 1. Which numerical methods may be combined together? 2. How can different numerical methods be matched together? In doing so the author presents a number of useful combinations, for instance, the combination of various FEMs, the combinations of FEM-FDM, REM-FEM, RGM-FDM, etc. The combined methods have many advantages over single methods: high accuracy of solutions, less CPU time, less computer storage, easy coupling with singularities as well as the complicated boundary conditions. Since coupling techniques are essential to combinations, various matching strategies among different methods are carefully discussed. The author provides the matching rules so that optimal convergence, even superconvergence, and optimal stability can be achieved, and also warns of the matching pitfalls to avoid. Audience: The book is intended for both mathematicians and engineers and may be used as text for advanced students.

For a numerical method, its stability is a crucial issue in the sense that an unstable algorithm can render it useless in practical computation. For an over-determined linear algebraic equation , its stability is typically evaluated by using the traditional condition number, Cond= , where and are the maximal and the minimal singular values of matrix , respectively. In this book, the concept of an effective condition number, Cond_eff= , is introduced. Cond_eff is smaller, and generally much smaller, than Cond, and is a better stability criterion. The Cond_eff can be used as an estimation on stability for the numerical solutions of partial differential equations (PDEs) using algorithms, such as the collocation Trefftz method, the spectral method, the finite difference method, and the finite element method. An analysis of stability integrated with errors is the focus of this book.