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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.
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.
The purpose of this book is to present the current state of the art
of the Virtual Element Method (VEM) by collecting contributions
from many of the most active researchers in this field and covering
a broad range of topics: from the mathematical foundation to real
life computational applications. The book is naturally divided into
three parts. The first part of the book presents recent advances in
theoretical and computational aspects of VEMs, discussing the
generality of the meshes suitable to the VEM, the implementation of
the VEM for linear and nonlinear PDEs, and the construction of
discrete hessian complexes. The second part of the volume discusses
Virtual Element discretization of paradigmatic linear and
non-linear partial differential problems from computational
mechanics, fluid dynamics, and wave propagation phenomena. Finally,
the third part contains challenging applications such as the
modeling of materials with fractures, magneto-hydrodynamics
phenomena and contact solid mechanics. The book is intended for
graduate students and researchers in mathematics and engineering
fields, interested in learning novel numerical techniques for the
solution of partial differential equations. It may as well serve as
useful reference material for numerical analysts practitioners of
the field.
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