ABSTRACT
In this chapter the authors present the details of the mathematics of space-time finite element method, space-time strip, and time-marching. They consider space-time finite element methods based on various methods of approximation for non-self-adjoint as well as non-linear space-time differential operators for 1D, 2D, and 3D initial value problems (IVPs). In space-time Galerkin method (STGM), space-time Petrov-Galerkin method (STPGM), and space-time weighted residual method (STWRM), the authors begin by constructing a space-time integral form based on fundamental lemma using the mathematical model describing the IVP. The authors then consider model problems describing evolutions that contain non-self-adjoint and non-linear space-time operators. Model problems consist of: 1D scalar wave equation, 1D pure advection, 1D transient convection-diffusion equation, 1D transient Burgers equation, 1D transient diffusion-reaction equation, 1D normal shocks in Riemann shock tube, and 2D liquid-solid and solid-liquid phase transition. For the model problems formulations are presented using STGM, STGM/WF, and space-time least squares process (STLSP) based on residual functional.
