Graded Finite Element Methods for Elliptic Problems in Nonsmooth Domains

This book presents graded finite element methods for elliptic problems in nonsmooth domains with mixed boundary conditions. Meant to be introductory and self-contained, it covers basic mathematical theory and numerical tools that address major challenges imposed by the singular solution. Since Kondratiev’s 1967 seminal paper [71], there has been significant progress on the mathematical analysis of singular problems. One purpose of this book is to synthesize the relevant literature, and introduce important theoretical results and their applications to a broader audience in the computation community. With a systematic approach, this book demonstrates the development of graded mesh finite element techniques for singular solutions, in which concepts in several areas (e.g., functional analysis, Sobolev space theory, numerical analysis, and scientific computing) play a critical role.