This work provides an overview of a posteriori error assessment techniques for Finite Element (FE) based numerical models. These tools aim at estimating and controlling the discretization error in scientific computational models, being the basis for the numerical verification of the FE solutions. The text discusses the capabilities and limitations of classical methods to build error estimates which can be used to control the quality of numerical simulations and drive adaptive algorithms, with a focus on Computational Mechanics engineering applications. Fundamentals principles of residual methods, smoothing (recovery) methods, and constitutive relation error (duality based) methods are thus addressed along the manuscript. Attention is paid to recent advances and forthcoming research challenges on related topics. The book constitutes a useful guide for students, researchers, or engineers wishing to acquire insights into state-of-the-art techniques for numerical verification.

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This second volume focuses on applications in engineering, biomedical engineering, computational physics and computer science.

This monograph provides a compendium of established and novel error estimation procedures applied in the field of Computational Mechanics. It also includes detailed derivations of these procedures to offer insights into the concepts used to control the errors obtained from employing Galerkin methods in finite and linearized hyperelasticity. The Galerkin methods introduced are considered advanced methods because they remedy certain shortcomings of the well-established finite element method, which is the archetypal Galerkin (mesh-based) method. In particular, this monograph focuses on the systematical derivation of the shape functions used to construct both Galerkin mesh-based and meshfree methods. The mesh-based methods considered are the (conventional) displacement-based, (dual-)mixed, smoothed, and extended finite element methods. In addition, it introduces the element-free Galerkin and reproducing kernel particle methods as representatives of a class of Galerkin meshfree methods. Including illustrative numerical examples relevant to engineering with an emphasis on elastic fracture mechanics problems, this monograph is intended for students, researchers, and practitioners aiming to increase the reliability of their numerical simulations and wanting to better grasp the concepts of Galerkin methods and associated error estimation procedures.

In the past one hundred years, two major realities have changed both science and religion. The world of science has been enriched by quantum physics, the computation of the age of the universe, archaeological data in the Middle East, and a scientific stress on historical writing. The world of religion has been enriched by the establishment of the World Council of Churches and the Second Vatican Council. In the past fifty years, major scientists and major religious leaders have met together again and again. In the past fifty years, religious leaders from Christianity, Islam, and Judaism have held a number of thought-provoking conferences. In this volume, these gatherings are reviewed and evaluated. Two major religious problems have challenged the science-religion discussions, namely, which God should the scientists agree on, the Trinitarian God, Allah, or Yahweh? Which history of the universe sponsored by these three religions should scientists be looking for? This volume raises questions and suggests some preliminary forms of serious discussion.

In 1848 James Challis showed that smooth solutions to the compressible Euler equations can become multivalued, thus signifying the onset of a shock singularity. Today it is known that, for many hyperbolic systems, such singularities often develop. However, most shock-formation results have been proved only in one spatial dimension. Serge Alinhac's groundbreaking work on wave equations in the late 1990s was the first to treat more than one spatial dimension. In 2007, for the compressible Euler equations in vorticity-free regions, Demetrios Christodoulou remarkably sharpened Alinhac's results and gave a complete description of shock formation. In this monograph, Christodoulou's framework is extended to two classes of wave equations in three spatial dimensions. It is shown that if the nonlinear terms fail to satisfy the null condition, then for small data, shocks are the only possible singularities that can develop. Moreover, the author exhibits an open set of small data whose solutions form a shock, and he provides a sharp description of the blow-up. These results yield a sharp converse of the fundamental result of Christodoulou and Klainerman, who showed that small-data solutions are global when the null condition is satisfied. Readers who master the material will have acquired tools on the cutting edge of PDEs, fluid mechanics, hyperbolic conservation laws, wave equations, and geometric analysis.

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