This book discusses numerical methods for solving time-fractional evolution equations. The approach is based on first discretizing in the spatial variables by the Galerkin finite element method, using piecewise linear trial functions, and then applying suitable time stepping schemes, of the type either convolution quadrature or finite difference. The main concern is on stability and error analysis of approximate solutions, efficient implementation and qualitative properties, under various regularity assumptions on the problem data, using tools from semigroup theory and Laplace transform. The book provides a comprehensive survey on the present ideas and methods of analysis, and it covers most important topics in this active area of research. It is recommended for graduate students and researchers in applied and computational mathematics, particularly numerical analysis.

Existence, Uniqueness, and Regularity of Solutions.- Semidiscrete Discretization.- Convolution Quadrature.- Finite Difference Methods: Construction and Implementation.- Finite Difference Methods on Uniform Meshes.- Finite Difference Methods on Graded Meshes.- Nonnegativity Preservation.- Discrete Fractional Maximal Regularity.- Subdiffusion with time-dependent coefficients.- Semilinear Subdiffusion Equations.- Time-Space Formulation and Finite Element Approximation.- A Spectral Petrov-Galerkin Method.- Incomplete Iterative Solution at the Time Levels.- Optimal Control with Subdiffusion Constraint.- Backward Subdiffusion Problems.- Appendix: Mathematical Preliminaries.