c2000, Joseph E. Flaherty, all rights reserved. These notes are intended for classroom use by Rensselaer students taking courses CSCI, MATH Copy

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1 CSCI, MATH 6860 FINITE ELEMENT ANALYSIS Lecture Notes: Spring 2000 Joseph E. Flaherty Amos Eaton Professor Department of Computer Science Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, New York 12180

2 c2000, Joseph E. Flaherty, all rights reserved. These notes are intended for classroom use by Rensselaer students taking courses CSCI, MATH Copying or downloading by others for personal use is acceptable with notication of the author. ii

3 CSCI, MATH 6860: Finite Element Analysis Spring 2000 Outline 1. Introduction 1.1. Historical Perspective 1.2. Weighted Residual Methods 1.3. A Simple Finite Element Problem 2. One-Dimensional Finite Element Methods 2.1. Introduction 2.2. Galerkin's Method and Extremal Principles 2.3. Essential and Natural Boundary Conditions 2.4. Piecewise Lagrange Approximation 2.5. Hierarchical Bases 2.6. Interpolation Errors 3. Multi-Dimensional Variational Principles 3.1. Galerkin's Method and Extremal Principles 3.2. Function Spaces and Approximation 3.3. Overview of the Finite Element Method 4. Finite Element Approximation 4.1. Introduction 4.2. Lagrange Bases on Triangles 4.3. Lagrange Bases on Rectangles 4.4. Hierarchical Bases 4.5. Three-dimensional Bases 4.6. Interpolation Errors 5. Mesh Generation and Assembly 5.1. Introduction iii

4 5.2. Mesh Generation 5.3. Data Structures 5.4. Coordinate Transformations 5.5. Generation of Element Matrices and Their Assembly 5.6. Assembly of Vector Systems 6. Numerical Integration 6.1. Introduction 6.2. One-Dimensional Gaussian Quadrature 6.3. Multi-Dimensional Gaussian Quadrature 7. Discretization Errors 7.1. Introduction 7.2. Convergence and Optimality 7.3. Perturbations 8. Adaptivity 8.1. Introduction 8.2. h-renement 8.3. p- and hp-renement 9. Parabolic Problems 9.1. Introduction 9.2. Semi-Discrete Galerkin Methods: The Method of Lines 9.3. Finite Element Methods in Time 9.4. Convergence and Stability 9.5. Convection-Diusion Systems 10. Hyperbolic Problems Introduction Flow Problems and Upwind Weighting Articial Diusion iv

5 10.4. Streamline Weighting 11. Linear Systems Solution Introduction Banded Gaussian Elimination and Prole Techniques Nested Dissection and Domain Decomposition Conjugate Gradient Methods Nonlinear Problems and Newton's Method v

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7 Bibliography [1] A.K. Aziz, editor. The Mathematical Foundations of the Finite Element Method with Applications to Partial Dierential Equations, New York, Academic Press. [2] I. Babuska, J. Chandra, and J.E. Flaherty, editors. Adaptive Computational Methods for Partial Dierential Equations, Philadelphia, SIAM. [3] I. Babuska, O.C. Zienkiewicz, J. Gago, and E.R. de A. Oliveira, editors. Accuracy Estimates and Adaptive Renements in Finite Element Computations. John Wiley and Sons, Chichester, [4] K.-J. Bathe. Finite Element Procedures. Prentice Hall, Englewood Clis, [5] E.B. Becker, G.F. Carey, and J.T. Oden. Finite Elements: An Introduction, volume I. Prentice Hall, Englewood Clis, [6] M.W. Bern, J.E. Flaherty, and M. Luskin, editors. Grid Generation and Adaptive Algorithms, volume 113 of The IMA Volumes in Mathematics and its Applications, New York, Springer. [7] C.A. Brebia. The Boundary Element Method for Engineers. Pentech Press, London, second edition, [8] S.C. Brenner and L.R. Scott. The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, [9] G.F. Carey. Computational Grids: Generation, Adaptation, and Solution Strategies. Series in Computational and Physical Processes in Mechanics and Thermal science. Taylor and Francis, New York, [10] G.F. Carey and J.T. Oden. Finite Elements: A Second Course, volume II. Prentice Hall, Englewood Clis, [11] G.F. Carey and J.T. Oden. Finite Elements: Computational Aspects, volume III. Prentice Hall, Englewood Clis, vii

8 [12] P.G. Ciarlet. The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, [13] K. Clark, J.E. Flaherty, and M.S. Shephard, editors. Applied Numerical Mathematics, volume 14, Special Issue on Adaptive Methods for Partial Dierential Equations. [14] R.D. Cook, D.S. Malkus, and M.E. Plesha. Concepts and Applications of Finite Element Analysis. John Wiley and Sons, New York, third edition, [15] K. Eriksson, D. Estep, P. Hansbo, and C. Johnson. Computational Dierential Equation. Cambridge, Cambridge, [16] G. Fairweather. Finite Element Methods for Dierential Equations. Marcel Dekker, Basel, [17] B. Finlayson. The Method of Weighted Residuals and Variational Principles. Academic Press, New York, [18] J.E. Flaherty, P.J. Paslow, M.S. Shephard, and J.D. Vasilakis, editors. Adaptive methods for Partial Dierential Equations, Philadelphia, SIAM. [19] R.H. Gallagher, J.T. Oden, C. Taylor, and O.C. Zienkiewicz, editors. Finite Elements in Fluids: Mathematical Foundations, Aerodynamics and Lubrication, volume 2, London, John Wiley and Sons. [20] R.H. Gallagher, J.T. Oden, C. Taylor, and O.C. Zienkiewicz, editors. Finite Elements in Fluids: Viscous Flow and Hydrodynamics, volume 1, London, John Wiley and Sons. [21] R.H. Gallagher, O.C. Zienkiewicz, J.T. Oden, M. Morandi Cecchi, and C. Taylor, editors. Finite Elements in Fluids, volume 3, London, John Wiley and Sons. [22] V. Girault and P.A. Raviart. Finite Element Approximations of the Navier-Stokes Equations. Number 749 in Lecture Notes in Mathematics. Springer-Verlag, Berlin, [23] T.J.R. Hughes, editor. Finite Element Methods for Convection Dominated Flows, volume 34 of AMD, New York, ASME. [24] T.J.R. Hughes. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice Hall, Englewood Clis, viii

9 [25] B.M. Irons and S. Ahmed. Techniques of Finite Elements. Ellis Horwood, London, [26] C. Johnson. Numerical Solution of Partial Dierential Equations by the Finite Element method. Cambridge, Cambridge, [27] N. Kikuchi. Finite Element Methods in Mechanics. Cambridge, Cambridge, [28] Y.W. Kwon and H. Bang. The Finite Element Method Using Matlab. CRC Mechanical Engineering Series. CRC, Boca Raton, [29] L. Lapidus and G.F. Pinder. Numerical Solution of Partial Dierential Equations in Science and Engineering. Wiley-Interscience, New York, [30] D.L. Logan. A First Course in the Finite Element Method using ALGOR. PWS, Boston, [31] J.T. Oden. Finite Elements of Nonlinear Continua. Mc Graw-Hill, New York, [32] J.T. Oden and G.F. Carey. Finite Elements: Mathematical Aspects, volume IV. Prentice Hall, Englewood Clis, [33] D.R.J. Owen and E. Hinton. Finite Elements in Plasticity-Theory and Practice. Pineridge, Swansea, [34] D.D. Reddy and B.D. Reddy. Introductory Functional Analysis: With Applications to Boundary Value Problems and Finite Elements. Number 27 in Texts in Applied Mathematics. Springer-Verlag, Berlin, [35] J.N. Reddy. The Finite Element Method in Heat Transfer and Fluid Dynamics. CRC, Boca Raton, [36] C. Schwab. P- And Hp- Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Numerical Mathematics and Scientic Computation. Clarendon, London, [37] G. Strang and G. Fix. Analysis of the Finite Element Method. Prentice-Hall, Englewood Clis, [38] B. Szabo and I. Babuska. Finite Element Analysis. John Wiley and Sons, New York, [39] F. Thomasset. Implementation of Finite Element Methods for Navier-Stokes Equations. Springer Series in Computational Physics. Springer-Verlag, New York, ix

10 [40] V. Thomee. Galerkin Finite Element Methods for Parabolic Problems. Number 1054 in Lecture Notes in Mathematics. Springer-Verlag, Berlin, [41] R. Verfurth. A Review of Posteriori Error Estimation and Adaptive Mesh- Renement Techniques. Teubner-Wiley, Stuttgart, [42] R. Vichevetsky. Computer Methods for Partial Dierential Equations: Elliptic Equations and the Finite-Element Method, volume 1. Prentice-Hall, Englewood Clis, [43] R. Wait and A.R. Mitchell. The Finite Element Analysis and Applications. John Wiley and Sons, Chichester, [44] R.E. White. An Introduction to the Finite Element Method with Applications to Nonlinear Problems. John Wiley and Sons, New York, [45] J.R. Whiteman, editor. The Mathematics of Finite Elements and Applications V, MAFELAP 1984, London, Academic Press. [46] J.R. Whiteman, editor. The Mathematics of Finite Elements and Applications VI, MAFELAP 1987, London, Academic Press. [47] O.C. Zienkiewicz. The Finite Element Method. Mc Graw-Hill, New York, third edition, [48] O.C. Zienkiewicz and R.L. Taylor. Finite Element Method: Solid and Fluid Mechanics Dynamics and Non-Linearity. Mc Graw-Hill, New York, x