The Mimetic Finite Difference Method for Elliptic Problems

Description

This book describes the theoretical and computational aspects of the mimetic finite difference method for a wide class of multidimensional elliptic problems, which includes diffusion, advection-diffusion, Stokes, elasticity, magnetostatics and plate bending problems. The modern mimetic discretization technology developed in part by the Authors allows one to solve these equations on unstructured polygonal, polyhedral and generalized polyhedral meshes. The book provides a practical guide for those scientists and engineers that are interested in the computational properties of the mimetic finite difference method such as the accuracy, stability, robustness, and efficiency. Many examples are provided to help the reader to understand and implement this method. This monograph also provides the essential background material and describes basic mathematical tools required to develop further the mimetic discretization technology and to extend it to various applications Front Matter....Pages i-xvi Front Matter....Pages 1-1 Model elliptic problems....Pages 3-40 Foundations of mimetic finite difference method....Pages 41-65 Mimetic inner products and reconstruction operators....Pages 67-89 Mimetic discretization of bilinear forms....Pages 91-113 Front Matter....Pages 115-115 The diffusion problem in mixed form....Pages 117-154 The diffusion problem in primal form....Pages 155-195 Maxwell’s equations....Pages 197-219 The Stokes problem....Pages 221-260 Front Matter....Pages 261-261 Elasticity and plates....Pages 263-287 Other linear and nonlinear mimetic schemes....Pages 289-310 Analysis of parameters and maximum principles....Pages 311-337 Diffusion problem on generalized polyhedral meshes....Pages 339-370 Back Matter....Pages 371-394