Parallel space-time discretisation methods

Abstract

The main aim of this doctoral thesis is to employ parallel methods to solve parabolic partial differential equations, specifically those leading to transient heat (diffusion) equations. One method for solving these problems is through the use of the semi-discrete finite element method. This scheme involves first discretising the spatial domain using the finite element method, which results in a system of ordinary differential equations. Next, this system is discretised over time using a time-stepping scheme, such as the Euler or Crank-Nicolson method. The resulting semi-discrete problem can then be solved in parallel using the Parareal method, which offers a relatively straightforward approach. The author of this thesis proposes a novel combination of the Parareal algorithm and a domain decomposition method based on the Schur complement approximation to increase the parallelism. This domain decomposition allows for the concurrent solutions of the spatial subproblems within each time slice.

Another method for solving parabolic partial differential equations is to use the finite element method directly on the space-time domain. This approach does not require any underlying tensor structure as in the previous case and allows for the use of general unstructured finite elements. However, in the final part of the doctoral thesis, the so-called Fast Diagonalisation Method is discussed, which uses the tensor-product technique to divide the space-time domain into a system of linear equations along the time interval and the system associated with the spatial domain. This method provides a parallel scheme along the time interval using the eigenvalues of the linear system in time. As the eigenvalues are complex, the author proposes a novel combination of the Fast Diagonalisation Method with the Preconditioning for REal matrices with Square Blocks (PRESB) method. The PRESB method utilises the complex structure of the obtained spatial systems to construct an efficient preconditioner.

The main contributions of this doctoral thesis are as follows: 1. A more in-depth examination of the Main Theorem of the well-posedness of a weak formulation for a parabolic problem, which Eberhard H. E. Zeidler has established in his work.

2. An overview of the Parareal method, including a discussion of potential implementation options and the proposal of a novel combination of the Parareal with the DDM based on the Schur complement approximation.

3. A summary of the space-time finite element method and the proposal of a novel combination of the Fast Diagonalisation Method and the PRESB algorithm (along with FGMRES method and multigrid), which demonstrates a potential of a full parallelisation of the space-time problem.