Space-Time-Parallel Poroelasticity Simulation

The accurate, reliable and efficient numerical approximation of multi-physics processes in heterogeneous porous media with varying media coefficients that include fluid flow and structure interactions is of fundamental importance in energy, environmental, petroleum and biomedical engineering applications fields for instance. Important applications include subsurface compaction drive, carbon sequestration, hydraulic and thermal fracturing and oil recovery. Biomedical applications include the simulation of vibration therapy for osteoporosis processes of trabeculae bones, estimating stress levels induced by tumour growth within the brain or next-generation spinal disc prostheses. Variational space-time methods offers some appreciable advantages such as the flexibility of the triangulation for complex geometries in space and natural local time stepping, the straightforward construction of higher-order approximations and the application of efficient goal-oriented (duality-based) adaptivity concepts. In addition to that, uniform space-time variational methods appear to be advantageous for stability and a priori error analyses of the discrete schemes. Especially (high-order) discontinuous in time approaches appear to have favourable properties due to the weak application of the initial conditions. The development of monolithic multi-physics schemes, instead of iterative coupling methods between the physical problems, is a key component of the research to reduce the modeling error. Special emphasis is on the development of efficient multi-physics and multigrid preconditioning technologies and their implementation. The simulation software DTM++ is a modularised framework written in C++11 and builds on top of deal.II toolchains. The implementation allows parallel simulations from notebooks up to cluster scale

A parallel preconditioning technique for an all-at-once system from subdiffusion equations with variable time steps

Volterra subdiffusion problems with weakly singular kernel describe the dynamics of subdiffusion processes well.The graded $L1$ scheme is often chosen to discretize such problems since it can handle the singularity of the solution near $t = 0$. In this paper, we propose a modification. We first split the time interval $[0, T]$ into $[0, T_0]$ and $[T_0, T]$, where $T_0$ ($0 < T_0 < T$) is reasonably small. Then, the graded $L1$ scheme is applied in $[0, T_0]$, while the uniform one is used in $[T_0, T]$. Our all-at-once system is derived based on this strategy. In order to solve the arising system efficiently, we split it into two subproblems and design two preconditioners. Some properties of these two preconditioners are also investigated. Moreover, we extend our method to solve semilinear subdiffusion problems. Numerical results are reported to show the efficiency of our method.Comment: 3 figures; 3 table