Solving the linear elasticity and Stokes equations by an optimal domain decomposition method derived algebraically involves the use of non standard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the non standard interface conditions are naturally defined at the boundary between elements. In this manuscript we present the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. An analysis of the boundary value problem with non standard condition is provided as well as the numerical evidence showing the advantages of the new method. Furthermore, we present and analyse a stabilisation method for the presented discretisation that allows the use of the same polynomial degrees for velocity and pressure discrete spaces. The original definition of the domain decomposition preconditioners is one-level, this is, the preconditioner is built only using the solution of local problems. This has the undesired consequence that the results are not scalable, it means that the number of iterations needed to reach convergence increases with the number of subdomains. This is the reason why we have also introduced, and tested numerically, two-level preconditioners. Such preconditioners use a coarse space in their construction. We consider two finite element discretisations, namely, the hybrid discontinuous Galerkin and Taylor-Hood discretisations for the nearly incompressible elasticity problems and Stokes equations.
|Date of Award||19 Jun 2017|
- University Of Strathclyde
|Sponsors||EPSRC (Engineering and Physical Sciences Research Council) & University of Strathclyde|
|Supervisor||Victorita Dolean Maini (Supervisor) & Gabriel Barrenechea (Supervisor)|