A Constructive Introduction to Finite Elements Method

Book summary

Summary of the book…

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Keywords

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Contents

1.Variational formulations
1.1. A 1D model problem
1.2. A 2D model problem (lapace eqution)
1.3. Some other boundary value problems
1.4. An eigenvalue problem
1.5. Problems

2. Finite elements method
2.1. F.E.M. for a 1D model problem
2.2. F.E.M. for a 2D model problem
2.3. “Stiffness” matrix and “load” vector
2.4. F.E.M. for an eigenvalue problem
2.5. Problems

3.Error analysis in F.E.M
3.1.An error estimate for F.E.M., 1D problems
3.2. An error estimate for F.E.M., 2D problems
3.3. Some other types of finite elements spaces
3.4. Problems

4. Direct and Iterative methods
4.1. Gaussian elimination
4.2. Iterative methods
4.3. Some minimization algorithms
4.4. Problems

5. Finite differences method
5.1. Existence and uniqueness of solutions of T.P.B.V.P.
5.2. A finite differences method
5.3. Eigenvalue problems
5.4. Problems

6. Appendices
6.1. Appendix A
6.2. Apendix B
6.3. Problems
6.4. Notations
6.5. Subject index

Chapter

Ch. 1 Introduction

Abstract

abstract ch 1 ???

[1] Ambrosetti, A., Rabinowitz, P.H., Dual Variational methods in cricitcal point theory and applications, J. Funct. Anal 14(1973), 349-381

[2] Apostol, T.M., Calculus, I., One-variable calculus, with an introduction to linear algebra, John Wiley & Sons, 1967; II Multi-variable calculus and linear algebra with apolications, 1969.

[3] Brenner, S.C., Scott, L.R., The mathematical theory of finite element methods, Springer0Verlag, New-York, 1994.

[4] Canuto, C., Huyssaini, M.Y., Quarteroni, A., Zang, T.A., Spectral methjods in fluid dynamics, Springer-Verlag, 1988.

[5] Ciarlet, P.G., The finite element method for elliptic problems, North Holland, 1978.

[6] Collatz, L., The numerical treatment of differential equations, Springer-Verlag, 1966.

[7] Courant, R., Hilbert, D., Methods of mathematical physics, New-York, Interscience Publishers, Inc. 1962.

[8] Davies, A.J., The finite element method, A first approach, Clarendon Press, Oxford, 1980.

[9] Elsgolts, L., Differential equations and the calculus of variation, Mir Publishers, Moscow, 1977.

[10] Ihlenburg, F., Finite element analysis of acoustic scattering, Springer Series in Applied Mathematics, vol. 132, Berlin, 1998.

[11] Johnson, C., Numerical solutions of partial differential equations by the finite element method, Cambridge University Press, 1987.

[12] Lanczos, C., The variational principles of mechanics, Fourth Edition, Dover Publications, New York, 1970.

[13] Michlin, S.G., Smolickij, K.L., Approximate methods for solution of differential and integral equations, Elsevier, 1967.

[14] Oden, J.T., Reddy, J.N., An introduction to the mathematical theory of finite elements, John Wiley & Sons, 1976.

[15] O’Neil, P.V., Advanced enginnering mathematica, 3rd ed., Wadsworth Publishing Company, Belmont, 1991 (International Student Edition).

[16] Roos, H.G., Stynes, M., Tobiska, L., Numerical methods for singularly perturbed differential equations-convection diffusion and flow problems, Springer-Verlag, 1996.

[17] Rudin, W., Priciples of mathematical analysis, McGraw-Hill, 1976.

[18] Sneddon, I., Elements of partial differential equations, Mc Graw-Hill, 20th Priting, 1986.

[19] Strang, G., Fix, G., An analysis of the finite element method, Prentice Hall, Englewood Cliffs, N.J., 1973.

[20] Straughan, B., The energy method, stability and nonlinear convection, Springer series in Applied Mathematics, vol., Berlin, 1992.

[21] Taylor, M.E., Partial differential equations I, Basic Theory, Springer, 1996.

[22]  Trim, D.W., Applied partial differential equations, PWS-Kent, Boston, 1990 (International Student Edition).

Chapter

Ch. 2 Preliminaries

Abstract

abstract ch2???

[1] Akin, J.E., Finite element analysis for undergraduates, Academic Press, 1986.

[2] Bathe, K.J., Finite element procedures in engineering analysis, Prentice Hall, Englewood Cliffs, 1982.

[3] Collatz, L., The numerical treatment of differential equations, 3rd, Springer-Verlag, Berlin, 1960.

[4] Club Modulef, A library of computer procedures for finite elements analysis, INRIA, Rocquencourt, France.

[5] Davis, A.J., The finite element method, A first Approach, Oxford University Press, 1980.

[6] Demmel, J.W., Applied numerical linear algebra, SIAM Philadelphia, 1997.

[7] Golub, G.H., Van Loan, C.F., Matrix computations, 3rd edition, The Johns Hopkins Univ. Press, Baltimore & London, 1996.

[8] Gustafson, K.E., Rao, D.K.M., Numerical range: the field of values of linear operators and matrices, Springer, 1996.

[9] Hughes, T.J.R., The finite element method, Prentice Hall, Englewood Cliffs, 1987.

[10] Issacson, E., Keller, H.B., Analysis of numerical methods, Hojn Wiley & Sons, 1966.

[11] Johnson, C., Numerical solutions of partial differential equations by the finite element method, Cambridge University Press, 1987.

[12] Micula, Gh., Micula, S., Handbook of splines, Kluwer Academic Publishers, 1999.

[13] Ottosen, N., Petersson, H., Introduction to finite element method, Prentice Hall, 1992.

[14] Prenter, P.M., Spline and variational methods, Wiley, New York – London, 1975.

[15] Reddy, J.N., An introduction to the finite element method, Mc Graw-Hill, 1984.

[16] Saleri, F., Programming implementation of finite slements, School on numerical simulation of P.D.E.: Methods, Algorithms, Applications, ICTP Trieste, 1996.

[17] Schults, M.H., Spline analysis, Prentice-Hall Series in Automatic computation, 1973.

[18] Segerlind, L.J., Applied finite element analysis, John Wiley & Sons, 1984.

[19] Strang, G., Linear algebra and its applications, Academic Press, 1976.

[20] Street, R.L., Analysis and solution of partial differential equations, Brooks/Colle Publishing Company, 1973.

[21] Wait, R., Mitchell, A.R., Finite element analysis and applications, John Wiley & Sons, 1985.

[22] Yu, Ch., Axelsson, O., An improved inverse iteration process for solving a sparse symmetric positive definite generalized eigenproblem, Report 9547 (Nov.1995) University of Nijmegen.

Chapter

Ch. 3 ??

Abstract

abstract ch. 3 ???

[1] Braes, D., Finite elements: Theory, Fase solvers & Applications in solid mechanics, Cambridge Univ. Press, Cambridge 1997.

[2] Brenner, S.C., Scott, L.R., The mathematical theory of FEM, Springer-Verlag, New York, 1994.

[3] Carey, G.F., Oden, J.T., Finite elements: A scond course, vol. II, Prentice-Hall, 1983.

[4] Ciarlet, P.G., The finite element method for elliptic equations, North Holland, Amsterdam, 1978.

[5] Ciarlet, P.G., Raviart, P.A., General Lagrange & Hermite interpolation in Rⁿ with applications to FEM, Arch. Rat. Mech. Anal., 46 (1972), pp. 177-199.

[6] Davies, A.J., The element method, A first approach, Oxford Univ. Press, 1980.

[7] Johnson, C., Numerical solutions of partial differential equations by f.e.m., Cambridge University Press, 1987.

[8] Petrila, T., Gheorghiu, C.I., Finite elements method & applications, Romanian Academy Publishing House, 1987.

[9] Schatz, A.H., An analysis of the f.e.m. for second order elliptic b.v.p., In: A.H., Schatz. V.T. Thomee and W.L. Wendland, Mathematical theory of finite & boundary elemnt methods, Birkhauser-Verlag, Basel, 1990.

[10] Szabo, B.S., Babuska, I., Finite element analysis, John Wiley, 1991.

Chapter

Ch. 4 Stochastic…???

Abstract

abstract ch. 4 ???

[1] Atkinson, K.A., An introduction to numerical analysis, John Wiley & Sons, 1978.

[2] Ciarlet, P.G., Introduction to numerical linear algebra and optimization, Cambridge Univ. Press, Cambridge, U.K., 1989.

[3] Conte, S.D., de Boor, C., Elementary numerical analysis, Mc Graw Hill, 1980 (International Student Edition).

[4] Demmel, J.W., Applied numerical linear algebra,  SIAM Philadelphia 1987.

[5] Deuflhard, P., Hohmann, A., Numerical analysis, A first course in Scientific Complutation, Walter & Gruyter, 1995.

[6] Golub, G.H., Van Loan, C.F., Matrix computations, 3rd edition, The Johns  Hopkins Univ. Press, Baltimore & London, 1996.

[7] greenbaum, A., Iterative methods for solving linear systems,  SIAM, Philadelphia, 1997.

[8] Gregory, J., Redmond, D., Introduction to numerical analysis, John & Bartlett Publishers, 1994.

[9] Isaacson, E., Keller, H.B., Analysis of numerical methods, John Wiley & Sons, 1996.

[10] Kineaid, D., Cheney, D., Numerical analysis, Brooke / Cole Publishing Company, 1990.

[11] Raltson, A., Rabinowitz, P., A first course in numerical analysis, McGraw – Hill Book Company, 1978.

[12] Schwartz, H.R., Numerical analysis, A comprehensive introduction, John Wiley & Sons, 1989.

[13] Stuart, A.M., Humphries, A.R., Dynamical systems and numerical analysis, Camb ridge Univ. Press, 1996.

[14] Trefethen, L.N., Approximation theory and numerical linear algebra, in algorithms for approximation II, J.C. Mason & M.G. Cox (Eds.), Chapman, London, 1990.

[15] Trefethen, L.N., Pseudospectra of matrices D.F. Griffths & G.A. Watson (Eds.), Numerical analysis 1991, Logman Scientific & Tehnical, Harlow, Essex, UK, 1992.

[16] Trefethen, L.N., Baw, D.III, Numerical Linear Algebra, SIAM, Philadelphia, 1997.

[17] Ueberhuber, Ch.W., Numerical Caomputation, 1,2: Methods, Software and analysis, Springer, 1997.

[18] Varga, R.S., Functional analysis & approximation theory in numerical analysis,  SIAM Philadelphia, 1991.

Chapter

Ch. 5 Stochastic…???

Abstract

abstract ch. 5 ???

[1] Asher, U., Mattheij, R.M.N., Russell, D., Numerical solution of boundary value problems for ordinary differential equations, Classics in Applied Mathematics 13, SIAM Philadelphia, 1995.

[2] Babuska, I., Osborn, J.E., Finite element Galerkin approximation of the eigenvalue and eigenvectors of selfadjoint problems, Math. Comp. 52 (1989) 275-297.

[3] Chatelin, F., Eigenvalue of matrices, Wiley, New York, 1993.

[4] Ciarlet, P.G., Raviart, P.A., Maximum principle and uniform convergence for the f.e.m., Computer methods in applied mechanics and eingineering, 2(1973), 17-31.

[5] Collatz, L., The numerical treatment of differential equations, Springer-Verlag, 1960.

[6] Gheorghiu, C.I., Pop, S.I., A modified Chevishev-Rau method for hydrodynamic stability problem. Proceedings of ICAOR 1996, Cluj-Napoca, Romania, vol. II, pp. 119-126.

[7] Isaacson, E., Keller, H.B., Analysis of numerical methods, John Wiley & Sons, 1966.

[8] Iserles, A., A first course in the numerical analysis of differential equations, Cambridge texts in applied mathematics, Cambridge Univ. Press, 1996.

[9] Orszag, S.A., Accurate solution of Orr-Sommerfeld stability equations, J. Fluid Mech. 50 (1971(, 689-703.

[10] Strang, G., Linear algebra and its applictions, Academic Press, 1976.

[11] Trefethen, L.N., Bau, D.III, Numerical linear algebra,  SIAM, Philadelphia, 1997.

[12] Wilkinson, J.H., The algebraic eigenvalue problem, Clarendon Press, Oxford, U.K., 1965.

Chapter

Ch. 6 Stochastic…???

Abstract

abstract ch. 6 ???

[1] Asher, U., Mattheij, R.M.M., Russell, R.D., Numerical solution of boundary value problems for ordinary differential equations, Prentice Hall, 1988.

[2] Brezis, H., Analyse fonctionnelle. Theorie et applications, Masson, Paris, 1992.

[3] Ciarlet, P.G., The finite element method for elliptic problems, North Holland, Amsterdam, 1978.

[4] Ciralet, P.G., Numerical analysis of the finite element method, Les Presses de l’Universite de Montreal, 1976.

[5] Dinca, G., Variational methods and applications, Bucharest 1980 (in Romanian, with an English abstract).

[6] Glowinski, R., Lions, J.L., Tremolieres, R., Analyse numerique des inequations  variationnelles, Tone 1 et 2, Bordas, Paris, 1976.

[7] Hilderbrandt, E.  Wienholtz, E., Constructive proofs of representations theorems in separable Hilbert space, Comm. Pure Appl. Math. 17(1964) 369-373.

[8] Lax, P.D., Milgram, A.N., Parabolic equations, Annals. of Math. Studies, no.33(1954) Princenton Univ. Press.

[9] Lions, J.L., Stampacchia, G., Variational inequalities, Comm. Pure Appl. Math. 20(1967) 493-519

[10] Petrila, T., Gheorghiu, C.I., The finite elements method and applications, Romanian Academy Publishing House, Bucharest, 1987 (in Romanian, with an English abstract).

[11] Temam, R., Notions sur l’Approximation des fonctions, Notes de Faculte de Sciences d;Orsay, 1970.

[12] Thomee, V., Galerkin finite element methods for parabolic problems, Springer Lecture Notes in Mathematics 1054, Berlin, Heidelberg, New York, 1984.

[13] Zeidler, E., Applied functional analysis, Applied Functional Analysis. Applications to Mathematical Physics. Springer Verlag 1995 (Applied Mathematical Sciences 108).

Cite this book as:

C.I. Gheorghiu, A Constructive Introduction to Finite Elements Method, Ed. Quo-Vadis, Cluj-Napoca, 1999.

Book Title

A Constructive Introduction to Finite Elements Method

Publisher

Quo Vadis

Print ISBN

973-99137-0-9

Google scholar

The book on google scholar.

1999

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