## The Mathematical Theory of Finite Element MethodsMathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scienti?c disciplines and a resurgence of interest in the modern as well as the cl- sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and to encourage the teaching of new courses. TAMwillpublishtextbookssuitableforuseinadvancedundergraduate and beginning graduate courses, and will complement the Applied Mat- matical Sciences (AMS) series, which will focus on advanced textbooks and research-level monographs. Pasadena, California J.E. Marsden Providence, Rhode Island L. Sirovich College Park, Maryland S.S. Antman Preface to the Third Edition This edition contains four new sections on the following topics: the BDDC domain decomposition preconditioner (Section 7.8), a convergent ad- tive algorithm (Section 9.5), interior penalty methods (Section 10.5) and 1 Poincar ́ e-Friedrichs inequalities for piecewise W functions (Section 10.6). |

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### Contents

V | 2 |

VI | 4 |

VII | 5 |

VIII | 8 |

IX | 10 |

X | 11 |

XI | 13 |

XII | 14 |

LXXIV | 166 |

LXXV | 171 |

LXXVI | 173 |

LXXVII | 176 |

LXXVIII | 180 |

LXXIX | 184 |

LXXX | 186 |

LXXXI | 192 |

XIII | 16 |

XIV | 20 |

XV | 24 |

XVI | 27 |

XVII | 30 |

XVIII | 33 |

XIX | 36 |

XX | 37 |

XXI | 41 |

XXII | 43 |

XXIII | 50 |

XXV | 52 |

XXVI | 53 |

XXVII | 56 |

XXVIII | 57 |

XXIX | 60 |

XXX | 61 |

XXXI | 65 |

XXXII | 66 |

XXXIII | 67 |

XXXIV | 70 |

XXXVI | 72 |

XXXVII | 73 |

XXXVIII | 76 |

XXXIX | 77 |

XL | 78 |

XLI | 82 |

XLII | 86 |

XLIII | 87 |

XLIV | 88 |

XLV | 90 |

XLVI | 91 |

XLVII | 94 |

XLIX | 97 |

L | 101 |

LI | 106 |

LII | 111 |

LIII | 114 |

LIV | 119 |

LV | 120 |

LVI | 124 |

LVII | 126 |

LVIII | 130 |

LX | 133 |

LXI | 135 |

LXII | 137 |

LXIII | 139 |

LXIV | 142 |

LXV | 145 |

LXVI | 147 |

LXVII | 149 |

LXVIII | 152 |

LXIX | 156 |

LXX | 158 |

LXXI | 160 |

LXXII | 162 |

LXXIII | 163 |

LXXXII | 198 |

LXXXIII | 202 |

LXXXIV | 206 |

LXXXV | 211 |

LXXXVI | 216 |

LXXXVIII | 219 |

LXXXIX | 221 |

XC | 225 |

XCI | 230 |

XCII | 232 |

XCIII | 236 |

XCIV | 239 |

XCV | 242 |

XCVI | 243 |

XCVII | 245 |

XCVIII | 248 |

XCIX | 250 |

C | 254 |

CI | 262 |

CII | 265 |

CIII | 267 |

CIV | 268 |

CV | 272 |

CVI | 273 |

CVII | 275 |

CVIII | 282 |

CIX | 287 |

CX | 290 |

CXI | 297 |

CXII | 304 |

CXIII | 312 |

CXV | 314 |

CXVI | 321 |

CXVII | 324 |

CXVIII | 328 |

CXIX | 332 |

CXX | 334 |

CXXI | 337 |

CXXII | 339 |

CXXIII | 342 |

CXXIV | 348 |

CXXV | 354 |

CXXVI | 356 |

CXXVIII | 360 |

CXXIX | 362 |

CXXX | 364 |

CXXXI | 367 |

CXXXII | 370 |

CXXXIII | 372 |

CXXXV | 374 |

CXXXVI | 377 |

CXXXVII | 380 |

CXXXVIII | 381 |

CXXXIX | 384 |

394 | |

### Common terms and phrases

algorithm approximation assume Banach space bilinear form boundary conditions bounded Cauchy-Schwarz inequality Chapter coercive constant C depends convergence Corollary deﬁned deﬁnition denote diam dimensions Dirichlet Dirichlet boundary conditions edge elliptic equation equivalent error estimates example exercise exists a positive ﬁnd Finite Element ﬁnite element space ﬁrst G Vh given Hilbert space Hint holds I F(v implies independent of h inner product integral interpolant L2 norm Lagrange Lemma Lipschitz mapping mesh midpoints multigrid nodal variables nodes non-degenerate norm Note operator penalty method piecewise polynomials polynomials of degree positive constant preconditioner Proposition proved the following quasi-uniform Remark result Riesz Representation Theorem satisﬁes satisfy Scott Sect smooth Sobolev spaces solving subspace Suppose symmetric T G T triangle triangle inequality unique solution vanishes variational formulation variational problem vector Vh(P Vv G V weak derivatives

### Popular passages

Page 385 - R. (2003): Adaptive finite element methods for differential equations. Lectures in Mathematics ETH Zurich. Birkhauser Verlag, Basel.

Page xii - This book develops the basic mathematical theory of the finite element method, the most widely used technique for engineering design and analysis. One purpose of this book is to formalize basic tools that are commonly used by researchers in the field but never published. It is intended primarily for mathematics graduate students and mathematically sophisticated engineers and scientists.

Page x - The first author would also like to thank the Alexander von Humboldt Foundation for supporting her visit to Germany in the Summer of 2007, during which the work on this edition was completed.