by Benqi Guo (University of Manitoba, Canada)

This book provides comprehensive knowledge and up-to-date developments of the p and h-p finite element methods. Introducing systematically the Jacobi-weighted Sobolev and Besov spaces, it establishes the approximation theory in the framework of these spaces in n dimensions. This is turn leads to the optimal convergence of the p and h-p finite element methods with quasi-uniform meshes in two dimensions for problems with smooth solutions and singular solutions on polygonal domains.

The book is based on the author's research on the p and h-p finite element methods over the past three decades. This includes the recently established approximation theory in Jacobi-weighted Sobolev and Besov spaces and rigorous proof of the optimal convergence of the p and h-p finite element method with quasi-uniform meshes for elliptic problems on polygonal domains. Indeed, these have now become the mathematical foundation of the high-order finite/boundary element method. In addition, the regularity theory in the countably Babuska-Guo-weighted Sobolev spaces, which the author established in the mid-1980s, provides a unique mathematical foundation for the h-p finite element method with geometric meshes and leads to the exponential rate of convergence for elliptic problems on polygonal domains.

• The Jacobi-Weighted Besov and Sobolev Spaces
• Approximation Theory in the Framework of Jacobi-Weighted Besov and Sobolev Spaces in One, Two and Three Dimensions
• The P and H-P of the Finite Element Method Version with Quasi-Uniform Meshes in Two Dimensions for Elliptic Problems with Smooth and Singular Solutions
• Optimal Convergence of the P and H-P Version for Problems on Polygonal Domains
• The H-P Version of the Finite Element Method with Geometric Meshes
• Regularity of Boundary Value Problems on Polygonal Domains in Countably Babuska–Guo-Weighted Sobolev Spaces
• Exponential Convergence of the H-P Version on Geometric Meshes
• Applications to Mechanical and Engineering Problems: Linear Elasticity, Stokes Flow and Heat Transfer
• Appendices:
  • The Quasi-Exactness and Partial Reiteration Theorems of Modified Interpolation Spaces
  • Polynomial Extensions on a Triangle and on a Square