Preconditioning of a hybridized discontinuous Galerkin finite element method for the Stokes equations

We present optimal preconditioners for a recently introduced hybridized discontinuous Galerkin finite element discretization of the Stokes equations. Typical of hybridized discontinuous Galerkin methods, the method has degrees-of-freedom that can be eliminated locally (cell-wise), thereby significantly reducing the size of the global problem. Although the linear system becomes more complex to analyze after static condensation of these element degrees-of-freedom, the pressure Schur complement of the original and reduced problem are the same. Using this fact, we prove spectral equivalence of this Schur complement to two simple matrices, which is then used to formulate optimal preconditioners for the statically condensed problem. Numerical simulations in two and three spatial dimensions demonstrate the good performance of the proposed preconditioners

On the efficient preconditioning of the Stokes equations in tight geometries

If the Stokes equations are properly discretized, it is well-known that the Schur complement matrix is spectrally equivalent to the identity matrix. Moreover, in the case of simple geometries, it is often observed that most of its eigenvalues are equal to one. These facts form the basis for the famous Uzawa and Krylov-Uzawa algorithms. However, in the case of complex geometries, the Schur complement matrix can become arbitrarily ill-conditioned having a significant portion of non-unit eigenvalues, which makes the established Uzawa preconditioner inefficient. In this article, we study the Schur complement formulation for the staggered finite-difference discretization of the Stokes problem in 3D CT images and synthetic 2D geometries. We numerically investigate the performance of the CG iterative method with the Uzawa and SIMPLE preconditioners and draw several conclusions. First, we show that in the case of low porosity, CG with the SIMPLE preconditioner converges faster to the discrete pressure and provides a more accurate calculation of sample permeability. Second, we show that an increase in the surface-to-volume ratio leads to an increase in the condition number of the Schur complement matrix, while the dependence is inverse for the Schur complement matrix preconditioned with the SIMPLE. As an explanation, we conjecture that the no-slip boundary conditions are the reason for non-unit eigenvalues of the Schur complement

Hybridised multigrid preconditioners for a compatible finite element dynamical core

Compatible finite element discretisations for the atmospheric equations of motion have recently attracted considerable interest. Semi-implicit timestepping methods require the repeated solution of a large saddle-point system of linear equations. Preconditioning this system is challenging since the velocity mass matrix is non-diagonal, leading to a dense Schur complement. Hybridisable discretisations overcome this issue: weakly enforcing continuity of the velocity field with Lagrange multipliers leads to a sparse system of equations, which has a similar structure to the pressure Schur complement in traditional approaches. We describe how the hybridised sparse system can be preconditioned with a non-nested two-level preconditioner. To solve the coarse system, we use the multigrid pressure solver that is employed in the approximate Schur complement method previously proposed by the some of the authors. Our approach significantly reduces the number of solver iterations. The method shows excellent performance and scales to large numbers of cores in the Met Office next-generation climate- and weather prediction model LFRic.Comment: 24 pages, 13 figures, 5 tables; accepted for publication in Quarterly Journal of the Royal Meteorological Societ

Formulation and analysis of a Schur complement method for fluid-structure interaction

This work presents a strongly coupled partitioned method for fluid-structure interaction (FSI) problems based on a monolithic formulation of the system which employs a Lagrange multiplier. We prove that both the semi-discrete and fully discrete formulations are well-posed. To derive a partitioned scheme, a Schur complement equation, which implicitly expresses the Lagrange multiplier and the fluid pressure in terms of the fluid velocity and structural displacement, is constructed based on the monolithic FSI system. Solving the Schur complement system at each time step allows for the decoupling of the fluid and structure subproblems, making the method non-iterative between subdomains. We investigate bounds for the condition number of the Schur complement matrix and present initial numerical results to demonstrate the performance of our approach, which attains the expected convergence rates.Comment: 27 pages, 4 figure

A block preconditioner for non-isothermal flow in porous media

In petroleum reservoir simulation, the industry standard preconditioner, the constrained pressure residual method (CPR), is a two-stage process which involves solving a restricted pressure system with Algebraic Multigrid (AMG). Initially designed for isothermal models, this approach is often used in the thermal case. However, it does not have a specific treatment of the additional energy conservation equation and temperature variable. We seek to develop preconditioners which better capture thermal effects such as heat diffusion. In order to study the effects of both pressure and temperature on fluid and heat flow, we consider a model of non-isothermal single phase flow through porous media. For this model, we develop a block preconditioner with an efficient Schur complement approximation. Both the pressure block and the approximate Schur complement are approximately inverted using an AMG V-cycle. The resulting solver is scalable with respect to problem size and parallelization.Comment: 35 pages, 3 figure

Effective Chorin–Temam algebraic splitting schemes for the steady Navier–stokes equations

This paper continues some recent work on the numerical solution of the steady incompressible Navier–Stokes equations. We present a new method, similar to the one presented in Rebholz et al., but with superior convergence and numerical properties. The method is efficient as it allows one to solve the same symmetric positive‐definite system for the pressure at each iteration, allowing for the simple preconditioning and the reuse of preconditioners. We also demonstrate how one can replace the Schur complement system with a diagonal matrix inversion while maintaining accuracy and convergence, at a small fraction of the numerical cost. Convergence is analyzed for Newton and Picard‐type algorithms, as well as for the Schur complement approximation