Fast solvers for elliptic PDEs form a pillar of scientific computing. They enable detailed and accurate simulations of electromagnetic fields, fluid flows, biochemical processes, and much more. This textbook provides an introduction to such solvers from the point of view of integral equation formulations, which lead to unparalleled accuracy and speed in many applications. The focus is on fast algorithms for handling the dense matrices that arise in the discretization of integral operators, such as the fast multipole method and fast direct solvers. The book also describes modern linear algebraic techniques that accelerate computations, such as randomized algorithms, interpolative decompositions, and data- sparse and rank-structured hierarchical matrix representations.
Fast Direct Solvers for Elliptic PDEs
- provides an accessible introduction to both integral equation methods for solving elliptic PDEs and fast multipole methods;
- is the first textbook to detail the active field of fast direct solvers, covering methods for both sparse and dense matrices; and
- is written with an emphasis on mathematical intuition, rather than theoretical details, illustrating concepts with numerous figures and detailed pseudocode for all key techniques.