Kronecker Product Approximation Preconditioners for Convection-diffusion Model Problems

We consider the iterative solution of the linear systems arising from four convection-diffusion model problems: the scalar convection-diffusion problem, Stokes problem, Oseen problem, and Navier-Stokes problem. We give the explicit Kronecker product structure of the coefficient matrices, especially the Kronecker product structure for the convection term. For the latter three model cases, the coefficient matrices have a $2 \times 2$ blocks, and each block is a Kronecker product or a summation of several Kronecker products. We use the Kronecker products and block structures to design the diagonal block preconditioner, the tridiagonal block preconditioner and the constraint preconditioner. We can find that the constraint preconditioner can be regarded as the modification of the tridiagonal block preconditioner and the diagonal block preconditioner based on the cell Reynolds number. That's the reason why the constraint preconditioner is usually better. We also give numerical examples to show the efficiency of this kind of Kronecker product approximation preconditioners

The INTERNODES method for applications in contact mechanics and dedicated preconditioning techniques

The mortar finite element method is a well-established method for the numerical solution of partial differential equations on domains displaying non-conforming interfaces. The method is known for its application in computational contact mechanics. However, its implementation remains challenging as it relies on geometrical projections and unconventional quadrature rules. The INTERNODES (INTERpolation for NOn-conforming DEcompositionS) method, instead, could overcome the implementation difficulties thanks to flexible interpolation techniques. Moreover, it was shown to be at least as accurate as the mortar method making it a very promising alternative for solving problems in contact mechanics. Unfortunately, in such situations the method requires solving a sequence of ill-conditioned linear systems. In this paper, preconditioning techniques are designed and implemented for the efficient solution of those linear systems

Micro- and macro-block factorizations for regularized saddle point systems

We present unique and existing micro-block and induced macro-block Crout-based factorizations for matrices from regularized saddle-point problems with semi-positive de¿nite regularization block. For the classical case of saddle-point problems we show that the induced macro-block factorizations mostly reduces to the factorization presented in [24]. The presented factorization can be used as a direct solution algorithm for regularized saddle-point problems as well as it can be used a basis for the construction of preconditioners

Alternative Solution Algorithms for Primal and Adjoint Incompressible Navier-Stokes

Regardless of the specific discretisation framework, the discrete incompressible Navier-Stokes equations present themselves in the form of a non-linear, saddle-point Oseentype system. Traditional CFD codes typically solve the system via the well-known SIMPLE-like algorithms, which are essentially block preconditioners based on Schur complement theory. Due to their “segregated” nature, which reduces to iteratively solving a sequence of linear systems smaller than the full Oseen and better conditioned, traditional SIMPLE-like algorithms have long been considered as the only viable strategy. However, recent progress in computational power and linear solver capabilities has led researchers to develop, for Oseen-type systems (and discrete Navier-Stokes in particular), a number of alternative preconditioners and solution schemes, found to be more efficient than SIMPLE-like strategies but previously deemed practically unfeasible in industrial contexts. The improved efficiency of novel preconditioners entails a) faster, more stable convergence and b) the possibility of driving residuals below more strict tolerances, which is sometimes difficult with SIMPLE due to stagnating behaviour. The second aspect in particular is extremely relevant in the context of adjoint-based optimisation, as evidence suggests that an adjoint system may be affected by convergence issues when the primal flow solution is not well converged. In this work, we present some solution schemes (both traditional and novel) implemented for the Mixed Hybrid Finite Volumes Navier-Stokes solver we introduced in our previous work. Performance, in terms of robustness and convergence properties, is assessed on a series of benchmark test cases. We also turn our attention to the discrete adjoint Navier-Stokes problem itself, which in essence requires solving a linear system similar to the original Oseen and therefore may benefit from the same preconditioning techniques. We show how the primal algorithms are adapted to the adjoint system, and we run a series of adjoint test cases to compare performance of various solution scheme

Deflation for the off-diagonal block in symmetric saddle point systems

Deflation techniques are typically used to shift isolated clusters of small eigenvalues in order to obtain a tighter distribution and a smaller condition number. Such changes induce a positive effect in the convergence behavior of Krylov subspace methods, which are among the most popular iterative solvers for large sparse linear systems. We develop a deflation strategy for symmetric saddle point matrices by taking advantage of their underlying block structure. The vectors used for deflation come from an elliptic singular value decomposition relying on the generalized Golub-Kahan bidiagonalization process. The block targeted by deflation is the off-diagonal one since it features a problematic singular value distribution for certain applications. One example is the Stokes flow in elongated channels, where the off-diagonal block has several small, isolated singular values, depending on the length of the channel. Applying deflation to specific parts of the saddle point system is important when using solvers such as CRAIG, which operates on individual blocks rather than the whole system. The theory is developed by extending the existing framework for deflating square matrices before applying a Krylov subspace method like MINRES. Numerical experiments confirm the merits of our strategy and lead to interesting questions about using approximate vectors for deflation.Comment: 26 pages, 12 figure

Spectral estimates and preconditioning for saddle point systems arising from optimization problems

In this thesis, we consider the problem of solving large and sparse linear systems of saddle point type stemming from optimization problems. The focus of the thesis is on iterative methods, and new preconditioning srategies are proposed, along with novel spectral estimtates for the matrices involved