Related ‘Classic’ Textbooks:

General Numerical Background:

Hornbeck, R. W., Numerical Methods, Quantum, 1975.

Westlake, J., Handbook of Numerical Matrix Inversion and Solution of Linear Equations, Krieger, 1975.

Abramowitz and Stegun: AMS55, Handbook of Mathematical Functions, National Bureau of Standards, 1964.

Wilkinson, J.H. and C. Reinsch, Linear Algebra, in Handbook for Automatic Computation, Vol. II, Ed. Bauer, et. al., Springer-Verlag, 1971

Finite Difference Methods:

Ames, W. F., Numerical Methods for Partial Differential Equations, Academic press, 1977, or latest edition.

Smith, G. D.,   Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford Univ Press, 3rd ed., 1985

Lapidus, L. and G. F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering, John Wiley and Sons, 1999 (paperback, same as 1982 hardback version).

Morton, K.W. and D.F. Mayers. Numerical Solution of Partial Differential Equations. Cambridge University Press, 1994.[Roughly equivalent to Smith.]

Mitchell, A.R. and D.F. Griffiths, Finite Difference Method in Partial Differential Equations, John Wiley & Sons, 1980.

Finite Element Methods:

Zienkiewicz, O. C., and R.L. Taylor, The Finite Element Method - Volumes 1, 2, and 3, 5th edition, Butterworth-Heinemann, 2000.

Segerlind, L. J., Applied Finite Element Analysis, J. Wiley and Sons, 2nd ed., 1984.

Lapidus, L. and G. F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering, John Wiley and Sons, 1999.

Strang, G. and G. J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, 1973.

Finlayson, B., Method of Weighted Residuals and Variational Principles, Academic Press, 1972.