General Texts in Numerical Analysis

K. Atkinson: An Introduction to Numerical Analysis, Wiley, (2nd ed.), 1989.
P.G. Ciarlet and J. L. Lions (eds), Handbook of Numerical Analysis,
     North Holland, 1990.
S. Conte and C. de Boor: Elementary Numerical Analysis, McGraw-Hill, 1980. 
G. Dahlquist and A. Bjorck: Numerical Methods, Prentice Hall, 1974.
P. Deuflhard and A. Hohmann, Numerical Analysis in Modern Scientific
      Computing, 2nd ed., Springer, 2003.
W. Gautschi: Numerical Analysis: an introduction, Birkhauser, 1997.
M. T. Heath, Scientific Computing: An Introductory Survey, 2nd ed.,
      McGraw-Hill, 2002.
E. Isaacson and H. Keller: Analysis of Numerical Methods, Wiley, 1966
     (or Dover 1994).
D. Kahaner, C. Moler, and S. Nash: Numerical Methods and Software,
     Prentice-Hall, 1989.
D. Kincaid and W. Cheney: Numerical Analysis: Mathematics of Scientific
     Computing, Brooks/Cole, 1996.
A. Quarteroni, R. Sacco, and F. Saleri: Numerical Mathematics, 2nd Edition,
     Springer, 2004.
A. Ralston and P. Rabinowitz: A First Course in Numerical Analysis, 
     McGraw-Hill, 1978.
L. Shampine, Allen, and Pruess: Fundamentals of Numerical Computing,
G. W. Stewart: Afternotes on Numerical Analysis, SIAM, 1996.
G. W. Stewart: Afternotes goes to Graduate School Lectures on Advanced
     Numerical Analysis, SIAM, 1998.
J. Stoer and R. Bulirsch: Introduction to Numerical Analysis, Springer-Verlag,
E. Suli and D. Mayers: An Introduction to Numerical Analysis, Cambridge, 2003.

Finite Element, Finite Volume, and Spectral Methods

O. Axelsson and V.A. Barker: Finite element solution of boundary value
     problems: theory and computation, Academic Press, 1984.
R. E. Bank: PLTMG A Software Package for Solving Elliptic Partial
     Differential Equations User's Guide 8.0, SIAM 1998.
M. Bernadou: Finite Element Methods for Thin Shell Problems, John Wiley,
D. Braess: Finite Elements: Theory, fast solvers, and applications in
     solid mechanics, Cambridge University Press, 2007, (3rd edition).
S. Brenner and L. R. Scott: The Mathematical Theory of Finite Element
     Methods, Springer-Verlag, 2008 (third edition).
D. Boffi, F. Brezzi, and M. Fortin, Mixed Finite Element Methods and
     Applications, Springer Series in Computational Mathematics, Vol 44, 2013.
F. Brezzi and M. Fortin: Mixed and Hybrid Finite Element Methods,
     Springer-Verlag, 1991.
P. G. Ciarlet:  The Finite Element Method for Elliptic Problems,
     North Holland, 1980.
P. G. Ciarlet and J. L. Lions (eds): Handbook of Numerical Analysis, Volumes
     I-VI, Elsevier.
K. Eriksson, D. Estep, P. Hansbo, C. Johnson: Computational Differential
     Equations, Cambridge University Press, 1996.
A. Ern and J-L. Guermond, Theory and Practice of Finite Elements,
     Springer-Verlag, 2004.
V. A. Girault and P-A. Raviart: Finite Element Methods for Navier-Stokes
     Equations, Springer-Verlag, 1986.
Mark S. Gockenbach: Understanding and Implementing the Finite Element
     Method, SIAM, 2006.
C. Grossmann, H-G. Roos, and M. Stynes: Numerical Treatment of
     Partial Differential Equations, Springer, 2007.
M. Gunzburger, Finite Element Methods for Viscous Incompressible Flows,
     Academic Press, 1989.
T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic
     Finite Element Analysis, Prentice Hall, 1987.
C. Johnson: Numerical Solution of Partial Differential Equations by
     the Finite Element Method, Dover Publications, 2009.
S. Larsson and V. Thomee, Partial Differential Equations with
     Numerical Methods, Springer, 2008.
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge
     University Press, 2002.
G. Strang and G. Fix: An Analysis of the Finite Element Method,
    Wellesley-Cambridg, 2nd edition, 2008.
B. Szab\'o and I. Babu\u ska: Finite Element Analysis
L. N. Trefethen: Spectral Methods in MATLAB, SIAM 2000.
O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method
     (4th edition), McGraw Hill, 1989.

Finite Difference Methods for Partial Differential Equations

S. K. Godunov and V. S. Ryabenki, Difference Schemes: an introduction
     to the underlying theory, North Holland, 1987.
B. Gustafsson, H-O. Kreiss, and J. Oliger, Time dependent problems and
     difference methods, Wiley, 1995.
C. Hall and T. Porsching: Numerical Analysis of Partial Differential
     Equations, Prentice Hall, 1990.
D. Kroner, Numerical Schemes for Conservation Laws, Wiley, 1997.
R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential
     Equations, Steady State and Time Dependent Problems, SIAM, 2007.
R. J. LeVeque, Numerical methods for conservation laws, Lectures in 
     Mathematics, ETH-Zurich, Birkhauser-Verlag, Basel, 2nd edition, 2005.
I. Iserles, A First Course in the Numerical Analysis of Differential
     Equations, Cambridge University Press, 1996.
A. Quarteroni and A. Valli, Numerical approximation of partial differential
     equations, Springer-Verlag, 1994.
R. D. Richtmyer and K. W. Morton:  Difference Methods for Initial Value
     Problems, Wiley-Interscience, 1967.
G. Sod: Numerical Methods in Fluid Mechanics: Initial and
     Initial Boundary Value Problems, Cambridge University Press, 1985.
J. Strikwerda: Finite Difference Schemes and Partial Differential
     Equations, Wadsworth and Brooks/Cole, 1989.
J. W. Thomas, Numerical Partial Differential Equations: Finite Difference
Methods, Springer-Verlag, 1995.

Multigrid and Domain Decomposition

J. H. Bramble: Multigrid methods, Pitman, 1993.
W. L. Briggs: A Multigrid Tutorial, SIAM, 1987.
W. Hackbusch: Multi-Grid Methods and Applications, Springer-Verlag, 1986.
B. Smith, P. Bjorstad, and W. Gropp: Domain Decomposition, Cambridge
     University Press, 1996.
J. Xu: Iterative methods by space decomposition and subspace corrections,
     SIAM Review 34 (1992), 581-613.


K. Atkinson and W. Han, Theoretical Numerical Analysis: A Functional
     Analysis Framework, Springer, 2001.  This book gives an introduction to
     functional analysis for graduate students pursuing research involving
     numerical analysis.
P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications,
     SIAM, 2013.
A. Tveito and R. Winther, Introduction to Partial Differential Equations:
     A Computational Approach.  Springer-Verlag, 1998.


T. Davis and K. Sigmon: MATLAB Primer, Seventh Edition, Chapman and Hall, 2004.
D. J. Higham and N. J. Higham: MATLAB Guide, Second Edition, SIAM, 2005.
C. Moler: Numerical Computing with MATLAB, SIAM, 2004.
A. Quarteroni and F. Saleri, Scientific Computing with Matlab, Springer, 2003.