Variational Space-Time Methods for the Elastic Wave Equation and the Diffusion Equation
Uwe Köcher · 2015
In this work we analyse variational space-time discretisation methods for an accurate and reliable numerical approximation of three-dimensional elastic ultrasonic waves in solids. This problem appears for instance in the field of the numerical simulation of structural health monitoring systems for modern fibre reinforced composites. For the discretisation in space, a discontinuous Galerkin method in the sense of a discontinuous Finite Element Method with arbitrary polynomial degree is presented. For the discretisation in time, various families of discontinuous, continuous and continuous differentiable Galerkin methods in the sense of time-marching schemes are presented. Special emphasis will be taken on the methods with higher-order polynomial approximation in the time domain. Moreover, we analyse variational space-time discretisation methods for a non-stationary diffusion equation. The diffusion equation appears here as simplified prototype for a general transport equation with diffusion processes, convection processes and chemical reactions. The diffusion equation is a non-stationary parabolic partial differential equation, with that a diffusive transport with a corresponding flux can be approximated. For the discretisation in space, a mixed Finite Element Method is presented. The applied mixed Finite Element Method simultaneously approximates here a scalar-valued primal variable and the corresponding vector-valued flux variable. This approach ensures a locally mass-conservative numerical approximation, which can not be done with the standard conforming Finite Element Method. Further, we analyse carefully the corresponding linear systems of the presented numerical schemes and study their numerical properties. The ill-conditioned linear systems, which appear especially for the higher-order in time approximations, can not be solved with standard numerical solvers. We present sophisticated iterative numerical solvers and derive efficient corresponding preconditioners for the efficient and reliable numerical solution of the ill-conditioned linear systems. The corresponding software is designed for high-performance parallel numerical simulations. We express the applied modern software-engineering aspects and the used parallel programming models as well as the used software libraries. The practical relevance of the presented variational space-time methods, the efficiency of the developed iterative solvers and the capabilities of the implemented software are exemplarily shown by various numerical experiments on high-performance workstations and a high-performance cluster system.