William F. Ames 
Numerical Methods for Partial Differential Equations [PDF ebook] 

This volume is designed as an introduction to the concepts of modern numerical analysis as they apply to partial differential equations. The book contains many practical problems and their solutions, but at the same time, strives to expose the pitfalls–such as overstability, consistency requirements, and the danger of extrapolation to nonlinear problems methods used on linear problems. Numerical Methods for Partial Differential Equations, Third Edition reflects the great accomplishments that have taken place in scientific computation in the fifteen years since the Second Edition was published. This new edition is a drastic revision of the previous one, with new material on boundary elements, spectral methods, the methods of lines, and invariant methods. At the same time, the new edition retains the self-contained nature of the older version, and shares the clarity of its exposition and the integrity of its presentation. – Material on finite elements and finite differences have been merged, and now constitute equal partners- Additional material has been added on boundary elements, spectral methods, the method of lines, and invariant methods- References have been updated, and reflect the additional material- Self-contained nature of the Second Edition has been maintained- Very suitable for PDE courses