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The key issues are a posteriori error estimation and it automatic mesh adaptation. Besides the traditional approach of energy-norm error control, a new duality-based technique, the Dual Weighted Residual method for goal-oriented error estimation, is discussed in detail. This method aims at economical computation of arbitrary quantities of physical interest by properly adapting the computational mesh. This is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters, and finally the stability of the resulting flow is investigated by solving an eigenvalue problem. 'Goal-oriented' adaptivity is designed to achieve these tasks with minimal cost.At the end of each chapter some exercises are posed in order to assist the interested reader in better understanding the concepts presented. Solutions and accompanying remarks are given in the Appendix.
Differential equations --- Finite element method. --- Numerical solutions.
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Ordinary differential equations --- Numerical solutions of differential equations --- Differential equations --- Equations différentielles --- Numerical solutions --- Solutions numériques --- Equations différentielles --- Solutions numériques --- 517.91 Differential equations --- 517.91 --- Numerical solutions. --- Monograph --- Numerical solutions&delete& --- Differential equations - Numerical solutions
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Differential equations, Elliptic --- Parallel algorithms --- Numerical solutions --- Data processing
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517.91 --- Differential inclusions --- Fuzzy sets --- Numerical solutions
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Differential equations, Partial --- Matrices --- Multigrid methods (Numerical analysis) --- Numerical solutions
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Differential equations, Partial --- Numerical solutions --- Data processing --- Diffpack (Computer file)
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519.62 --- 519.62 Numerical methods for solution of ordinary differential equations --- Numerical methods for solution of ordinary differential equations --- Delay differential equations --- Numerical analysis --- Numerical solutions --- Numerical solutions of differential equations --- Numerical solutions.
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As a partner to Volume 1: Dimensional Continuous Models, this monograph provides a self-contained introduction to algebro-geometric solutions of completely integrable, nonlinear, partial differential-difference equations, also known as soliton equations. The systems studied in this volume include the Toda lattice hierarchy, the Kac-van Moerbeke hierarchy, and the Ablowitz-Ladik hierarchy. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The theory presented includes trace formulas, algebro-geometric initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses basic techniques from the theory of difference equations and spectral analysis, some elements of algebraic geometry and especially, the theory of compact Riemann surfaces. The presentation is constructive and rigorous, with ample background material provided in various appendices. Detailed notes for each chapter, together with an exhaustive bibliography, enhance understanding of the main results.
Differential equations, Nonlinear --- Solitons --- Numerical solutions --- Solitons. --- Pulses, Solitary wave --- Solitary wave pulses --- Wave pulses, Solitary --- Connections (Mathematics) --- Nonlinear theories --- Wave-motion, Theory of --- Numerical analysis --- Numerical solutions. --- Differential equations, Nonlinear - Numerical solutions
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