Numerical Treatment of Partial Differential EquationsSpringer Science & Business Media, 11.08.2007 - 596 Seiten Many well-known models in the natural sciences and engineering, and today even in economics, depend on partial di?erential equations. Thus the e?cient numerical solution of such equations plays an ever-increasing role in state-- the-art technology. This demand and the computational power available from current computer hardware have together stimulated the rapid development of numerical methods for partial di?erential equations—a development that encompasses convergence analyses and implementational aspects of software packages. In 1988 we started work on the ?rst German edition of our book, which appeared in 1992. Our aim was to give students a textbook that contained the basic concepts and ideas behind most numerical methods for partial di?er- tial equations. The success of this ?rst edition and the second edition in 1994 encouraged us, ten years later, to write an almost completely new version, taking into account comments from colleagues and students and drawing on the enormous progress made in the numerical analysis of partial di?erential equations in recent times. The present English version slightly improves the third German edition of 2005: we have corrected some minor errors and added additional material and references. |
Inhalt
Weak Solutions | 125 |
The Finite Element Method | 173 |
Finite Element Methods for Unsteady Problems | 317 |
Singularly Perturbed Boundary Value Problems 375 | 376 |
Variational Inequalities Optimal Control | 436 |
Numerical Methods for Discretized Problems | 499 |
Textbooks and Monographs | 571 |
Index | 585 |
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Numerical Treatment of Partial Differential Equations Christian Grossmann,Hans-G. Roos,Martin Stynes Eingeschränkte Leseprobe - 2007 |
Häufige Begriffe und Wortgruppen
analysis applied approximation arbitrary associated Assume assumptions basis bilinear form boundary conditions boundary value problem bounded called Chapter choose computed Consequently consider consistency constant continuous convergence defined denote derivatives described difference differential equations discrete discrete problem discuss domain elliptic equivalent error error estimates estimate example Exercise exists Figure finite element method formulation functions Galerkin given gives grid Hence implies independent inequality initial integrals iteration layer leads Lemma linear linear system mapping matrix maximum mesh mixed norm numerical obtain operator optimal parabolic parameter penalty piecewise points polynomials positive principle Proof prove regularity Remark requires respect result satisfies scheme shows simple smooth solution solve space stability standard step technique Theorem tion triangle unique variational equation vector weak yields
Beliebte Passagen
Seite 370 - PROOF. The proof of this theorem is similar to the proof of Theorem 3.11.
Seite 130 - L2(flp(M)); it remains bilinear and positive definite, because as usual, in the definition of L2, functions that differ only on a set of measure zero are identified. Using the Hodge star operator *, we can introduce the codifferential operator...
Seite 210 - The advantage of the finite element method over the finite difference method is the ability to more accurately describe irregular system boundaries and heterogeneous properties.