Solving Numerical PDEs: Problems, Applications, Exercises
Springer Science & Business Media, Apr 5, 2012 - Mathematics - 434 pages
This book stems from the long standing teaching experience of the authors in the courses on Numerical Methods in Engineering and Numerical Methods for Partial Differential Equations given to undergraduate and graduate students of Politecnico di Milano (Italy), EPFL Lausanne (Switzerland), University of Bergamo (Italy) and Emory University (Atlanta, USA). It aims at introducing students to the numerical approximation of Partial Differential Equations (PDEs). One of the difficulties of this subject is to identify the right trade-off between theoretical concepts and their actual use in practice. With this collection of examples and exercises we try to address this issue by illustrating "academic" examples which focus on basic concepts of Numerical Analysis as well as problems derived from practical application which the student is encouraged to formalize in terms of PDEs, analyze and solve. The latter examples are derived from the experience of the authors in research project developed in collaboration with scientists of different fields (biology, medicine, etc.) and industry. We wanted this book to be useful both to readers more interested in the theoretical aspects and those more concerned with the numerical implementation.
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Part II Stationary Problems
3 Galerkinfinite element method for elliptic problems
4 Advectiondiffusionreaction ADR problems
Part III Time Dependent Problems
5 Equations of parabolic type
6 Equations of hyperbolic type
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algebraic approacimation assume backward Euler barycentric coordinates basis functions bilinear form boundary conditions boundary data Cauchy–Schwarz inequality coefficients coercivity column computed condition number constant continuous convergence corresponding defined degrees of freedom denote derivative diagonal differential equation Dirichlet conditions discretization error domain eigenvalues error Euler method exact solution fem1d finite differences Galerkin method given grid hand side initial condition int2d Th integral interpolation interval ISBN Lax-Milgram Lemma Lemma linear finite elements linear system mass matrix Mathematical analysis MATLAB matrix mesh nodes norm Numerical approximation Numerical results obtained oscillations parameters piecewise Poincaré inequality polynomial posedness preconditioner pressure previous exercise problem reads Program scheme Solution Exact Solution solve step symmetric term u2last un+1 Upwind values vector velocity verify vertexes weak formulation XSPAN