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The focus of this monograph is the development of space-time adaptive methods to solve the convection/reaction dominated non-stationary semi-linear advection diffusion reaction (ADR) equations with internal/boundary layers in an accurate and efficient way. After introducing the ADR equations and discontinuous Galerkin discretization, robust residual-based a posteriori error estimators in space and time are derived. The elliptic reconstruction technique is then utilized to derive the a posteriori error bounds for the fully discrete system and to obtain optimal orders of convergence. As coupled surface and subsurface flow over large space and time scales is described by (ADR) equation the methods described in this book are of high importance in many areas of Geosciences including oil and gas recovery, groundwater contamination and sustainable use of groundwater resources, storing greenhouse gases or radioactive waste in the subsurface.

Physical geography. --- Galerkin methods. --- Sinc-Galerkin methods --- Sinc methods --- Numerical analysis --- Numerical analysis. --- Differential equations, partial. --- Numerical Analysis. --- Partial Differential Equations. --- Geophysics/Geodesy. --- Geography --- Partial differential equations --- Mathematical analysis --- Partial differential equations. --- Geophysics. --- Geological physics --- Terrestrial physics --- Earth sciences --- Physics

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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.

Mechanics. --- Mechanics, Applied. --- Computer mathematics. --- Computational intelligence. --- Solid Mechanics. --- Computational Science and Engineering. --- Computational Intelligence. --- Intelligence, Computational --- Artificial intelligence --- Soft computing --- Computer mathematics --- Electronic data processing --- Mathematics --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Galerkin methods. --- Sinc-Galerkin methods --- Sinc methods --- Numerical analysis --- Solids. --- Data processing. --- Solid state physics --- Transparent solids

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This monograph presents numerical methods for solving transient wave equations (i.e. in time domain). More precisely, it provides an overview of continuous and discontinuous finite element methods for these equations, including their implementation in physical models, an extensive description of 2D and 3D elements with different shapes, such as prisms or pyramids, an analysis of the accuracy of the methods and the study of the Maxwell’s system and the important problem of its spurious free approximations. After recalling the classical models, i.e. acoustics, linear elastodynamics and electromagnetism and their variational formulations, the authors present a wide variety of finite elements of different shapes useful for the numerical resolution of wave equations. Then, they focus on the construction of efficient continuous and discontinuous Galerkin methods and study their accuracy by plane wave techniques and a priori error estimates. A chapter is devoted to the Maxwell’s system and the important problem of its spurious-free approximations. Treatment of unbounded domains by Absorbing Boundary Conditions (ABC) and Perfectly Matched Layers (PML) is described and analyzed in a separate chapter. The two last chapters deal with time approximation including local time-stepping and with the study of some complex models, i.e. acoustics in flow, gravity waves and vibrating thin plates. Throughout, emphasis is put on the accuracy and computational efficiency of the methods, with attention brought to their practical aspects. This monograph also covers in details the theoretical foundations and numerical analysis of these methods. As a result, this monograph will be of interest to practitioners, researchers, engineers and graduate students involved in the numerical simulation of waves.

Computer mathematics. --- Continuum physics. --- Applied mathematics. --- Mathematical Applications in the Physical Sciences. --- Classical Continuum Physics. --- Engineering. --- Mathematical physics. --- Physics. --- Engineering mathematics. --- Continuum mechanics. --- Appl.Mathematics/Computational Methods of Engineering. --- Numerical and Computational Physics. --- Continuum Mechanics and Mechanics of Materials. --- Computational Science and Engineering. --- Finite element method. --- Galerkin methods. --- Sinc-Galerkin methods --- Sinc methods --- Numerical analysis --- FEA (Numerical analysis) --- FEM (Numerical analysis) --- Finite element analysis --- Isogeometric analysis --- Mechanics. --- Mechanics, Applied. --- Computer science. --- Mathematical and Computational Engineering. --- Numerical and Computational Physics, Simulation. --- Classical and Continuum Physics. --- Solid Mechanics. --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Engineering --- Engineering analysis --- Mathematical analysis --- Informatics --- Science --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Mathematics --- Computer mathematics --- Electronic data processing --- Classical field theory --- Continuum physics --- Continuum mechanics --- Physical mathematics --- Natural philosophy --- Philosophy, Natural --- Physical sciences