Internal seminar of the SERENA team, Thursday 5 November, 4pm-5pm in building 13: Iain Smears: Robust and efficient preconditioners for the discontinuous Galerkin time-stepping method Abstract: The discontinuous Galerkin time-stepping method has many advantageous properties for solving parabolic equations. However, its practical use has been limited by the large and challenging nonsymmetric systems to be solved at each time-step. This work develops a fully robust and efficient preconditioning strategy for solving these systems. We first construct a left preconditioner, based on inf-sup theory, that transforms the linear system to a symmetric positive definite problem that can be solved by the preconditioned conjugate gradient (PCG) algorithm. We then prove that the transformed system can be further preconditioned by an ideal block diagonal preconditioner, leading to a condition number κ bounded by 4 for any time-step size, any approximation order and any positive self-adjoint spatial operators. Numerical experiments demonstrate the low condition numbers and fast convergence of the algorithm for both ideal and approximate preconditioners, and show the feasibility of the high-order solution of large problems.
Internal seminar of the SERENA team, Thursday 29 Octobre, 3pm-4pm in building 13: Sarah Ali Hassan: A posteriori error estimates for domain decomposition methods Abstract: This work develops tight a posteriori error estimates for the Domain Decomposition (DD) method with Robin transmission conditions in mixed finite element discretizations. An interface problem is solved iteratively where at each iteration local subdomain problems are solved, and information is then transferred to the neighboring subdomains. By estimating the error of the DD method as well as the discretization error, an effective criterion to stop the DD iterations is developed. The a posteriori estimates are based on the reconstruction techniques for pressures and fluxes.