Unlike most finite element books that cover time dependent processes (IVPs) in a cursory manner, The Finite Element Method for Initial Value Problems: Mathematics and Computations focuses on the mathematical details as well as applications of space-time coupled and space-time decoupled finite element methods for IVPs. Space-time operator classification, space-time methods of approximation, and space-time calculus of variations are used to establish unconditional stability of space-time methods during the evolution. Space-time decoupled methods are also presented with the same rigor. Stability of space-time decoupled methods, time integration of ODEs including the finite element method in time are presented in detail with applications. Modal basis, normal mode synthesis techniques, error estimation, and a posteriori error computations for space-time coupled as well as space-time decoupled methods are presented. This book is aimed at a second-semester graduate level course in FEM.
Introduction. Concepts from Functional Analysis and Calculus of Variations. Space-Time Coupled Classical Methods of Approximation. Space-Time Finite Element Method. Space-Time Decoupled or Quasi Finite Element Methods. Methods of Approximation for ODEs in Time. Finite Element for ODEs in Time. Stability Analysis of the Methods of Approximation. Mode Superposition Technique. Errors in Numerical Solutions of Initial Value Problems. Appendix A: Nondimensionalizing Mathematical Models. Appendix B: Mapping and Interpolation Theory. Appendix C: Numerical Integration using Gauss Quadrature. Index